What are the best theoretical or empirical defenses of equilibrium analysis in economics? An Anarchist FAQ (see Wikipedia page here) focuses its opposition to mainstream economics by criticizing equilibrium analysis. The FAQ notes that,

[Equilibrium analysis] is essentially a static tool used to analyse a dynamic system. It assumes stability where none exists. [...] Thus the whole concept is an unreal rather than valid abstraction of reality. Sadly, the notions of “perfect competition” and (Walrasian) “general equilibrium” are part and parcel of neoclassical economics. It attempts to show, in the words of Paul Ormerod, “that under certain assumptions the free market system would lead to an allocation of a given set of resources which was in a very particular and restricted sense optimal from the point of view of every individual and company in the economy.” [The Death of Economics, p. 45] This was what Walrasian general equilibrium proved. However, the assumptions required prove to be somewhat unrealistic (to understate the point).


[...] Given that general equilibrium theory does not satisfactorily encompass such things as profit, money, growth, instability or even firms, how it can be considered as even an adequate representation of any real capitalist economy is hard to understand. Yet, sadly, this perspective has dominated economics for over 100 years. There is almost no discussion of how scarce means are organised to yield outputs, the whole emphasis is on exchanges of ready made goods. This is unsurprising, as this allows economics to abstract from such key concepts as power, class and hierarchy.

As for the argument about using unrealistic assumptions in economic models, I'm aware that Milton Friedman had written a defense of that practice in his Essays in Positive Economics. However, I am unaware of any work addressing the specific critiques of equilibrium analysis cited in An Anarchist FAQ, or defending equilibrium analysis in economics from a philosophical, or empirical perspective.
There is a paper fully dedicated to this topic by Backhouse (2004).

The concept of equilibrium is often misunderstood by non-scientists and non-economists.  Following Robinson (1956: 57):


The word equilibrium, in ordinary speech, describes a relation between
bodies in space. The scales of a balance are in equilibrium when the balance is at rest. . . . If we are continually throwing coppers at random into
either scale, the balance is continually wobbling and never reaches equilibrium; but, at any moment, there is a definite equilibrium position
which it would quickly reach if, from that moment, we left it alone.

Hence, the equilibrium analysis can be readily applied to dynamic systems. Equilibrium analysis can be viewed as sort of thought experiment where to organize your thinking passage of time is held constant. Now thinking about equilibrium analysis in the way above still has few issues.
Again Robinson (1956: 59) writes:

Nor can we apply the metaphor of a balance which is seeking or tending
towards a position of equilibrium though prevented from actually reaching it through constant disturbances. In economic affairs the fact that
disturbances are known to be liable to occur makes expectations about
the future uncertain and has an important effect on any conduct (which,
in fact, is all economic conduct) directed towards future results. . . . A
belief that a particular share is going to rise in price causes people to
offer to buy it and so raises its price. . . . This element of ‘thinking makes
it so’ creates a situation where a cunning guesser who can guess what the
other guessers are going to guess is able to make a fortune. There are
then no solid weights to give us analogy with a pair of scales in balance.
The metaphor of equilibrium is treacherous. . .

However, this issue was later addressed as discipline evolved Backhouse (2004):

There is, however, an alternative way to think of equilibrium, not as the
static equilibrium analogous to that of a pendulum, but as inter-temporal
equilibrium where expectations are fulfilled (Milgate 1979, 1987, Phelps
1987). Because this idea is now so strongly associated with the New
Classical Macroeconomics and related theories, it is worth noting that Erik
Lindahl and his Swedish colleagues used the notion to attack the MisesHakek notion of neutral money. If expectations were correctly anticipated,
any rate of inflation was consistent with monetary equilibrium. The door
was opened to discretionary monetary policy. (see Laidler 1999: 58–60).

Furthermore, Backhouse (2004) also argues:

... This is the notion that equilibrium is where (a) agents’ decisions are compatible with each other and (b) no agent has any reason to
change his or her behavior. The second part of this condition (which might
be thought to encompass the first) is often represented by saying that agents
are maximizing utility subject to the constraints they face.

Consequently, in economics there are already several possible notions of equilibria:


Equilibrium as the absence of endogenous tendencies for change.
Equilibrium as balance of forces (using a mechanical analogy).
Equilibrium as correct expectations.
Equilibrium as meaning that no agent has any reason to change his
or her behavior.



We use the concepts above because they allows us to describe and analyze economy and economic relationship as well as to test theories we make about them. Following  Huw Dixon (1990: 356),


At its most general, we can say that ‘equilibrium’ is a method of solving
economic models. At a superficial level, an equilibrium is simply a
solution to a set of equations. ... However, there is more to it than that. Whilst economists rarely argue
over how to solve equations, they do argue over whether a particular
solution represents a ‘real’ equilibrium or not. What is at stake is the
economist’s view of economic agents and the market.

Next following further Backhouse (2004):

An economist who has argued the case for equilibrium analysis very forcefully is Robert Lucas (1980). He interprets equilibrium to mean individual
optimization and consistency of plans (market clearing though perfect
competition). The alternative is to use ‘free parameters’ – parameters that
are not based on individual optimizing behavior, such as those involved
in equations relating price changes to excess demand for commodities. His
reason is that we can never know whether free prarameters are reliable.
Equilibrium models, he claims, are reliable in that parameters are based on
microeconomic behavior which is observable and directly measurable in a
way in which free parameters are not. When economists had no means of
handling more complicated equilibrium models, it was necessary that they
work with free-parameter models in order to explain dynamics. Such work
was immensely productive.


... The substantive points in his argument are that
microeconomic behavior can be established by experimentation; that this
behavior can be described in terms of optimization; and (implicitly) that
aggregation is possible without the introduction of what he calls free
parameters. They define Lucas’s view of how economies work at the macroeconomic level. These are the points on which discussion should centre, not
the issue of equilibrium. This was nicely expressed by McCallum (1986:
412):

Whether or not one works with a model in which “all markets clear” is a
matter of convention, but whether this clearing pertains to auction-type
markets, or to ones with nominal contracts (perhaps implicit) that have
allocational effects, is not.
Another way to express Lucas’s support for equilibrium models and his
hostility toward free-parameter models is that he believes that it provides
explanations of why parameters are what they are. Free parameters are
unexplained, however strongly they are supported by data.


Hence to sum it up.

Equilibrium is far more wider and complex concept that commonly perceived.
It is a way how to put structure on raw data and how to understand mechanisms that generate the data we observe
It is useful concept even for empiricists as it helps us test different explanations for economic phenomena.


The arguments above respond to proper criticism of concept equilibrium in the philosophy of economics. There quoted passages from that anarchist website have far more problems and can be dismissed more easily.

[Equilibrium analysis] is essentially a static tool used to analyse a dynamic system. It assumes stability where none exists. [...] Thus the whole concept is an unreal rather than valid abstraction of reality.

This is incorrect prima facie.

You can use static tools to analyze dynamics systems if you do it at a given point in time.

For example, the whole concept of equilibrium is omnipresent in physics (see the famous Feynman Lectures on Physics). Physical systems such our solar system, earths crust etc are often analyzed using concept of static equilibrium (here is full lecture on this topics). Now I am fully aware Economics is not Physics, the point is that general statement made there is prima facie false assertion.
Static models can and are routinely applied to systems that in real life are ultimately dynamic (since time always marches forward all systems are). This is because we often on purpose want to analyze problem holding the passage of time constant as that makes problem more tractable. This is done in any field of science and there is no reason why it should not be applicable to economic problems. It would  be like arguing English can be used in one field but not another without providing any additional reasons.

The concept of equilibrium directly extend to dynamic analysis. As one can see in any modern micro or macro textbooks (e.g. see MWG Microeconomics Theory or Advanced Macroeconomics by Romer) economics often analyzes equilibria in dynamic fashion or explores dynamic properties of various equilibria.



However, the assumptions required prove to be somewhat unrealistic (to understate the point).



This was already addressed by the essay of Friedman which you cite. Models are not meant to be realistic if they would be they would cease to be models in the first place. Models simplify reality because it is beyond the capability of any human mind to comprehend whole reality at the same time. Good models will simplify reality in some aspects while be true to reality in others. One can criticize any model for being unrealistic and realistic in wrong places but arguing that 'realism' of a model per se, without any qualifications,  is relevant criterion to judge it is false.



There is almost no discussion of how scarce means are organised to yield outputs, the whole emphasis is on exchanges of ready made goods.



The above is literary not truth, as the discussion of how scarce means are organized to yield output is virtually omnipresent in economics (see sources cited above).
The criticism above is not exclusive, one could criticize sentence after sentence mentioned in that excerpt you posted in your question. Even if you would like to criticize equilibrium analysis most of that excerpt is not carefully constructed criticism but rather a discussion of some straw-man version of equilibrium/economic analysis.