Is time itself speeding up universally? Time moves more slowly near a mass than in relatively empty space.  For example, I get that an observer would see someone falling into a black hole to appear to move more slowly and get "stuck" on the event horizon, while the falling body would see the observer as aging more rapidly.

Additionally, an observer watching a passenger craft moving near the speed of light (high mass equivalent) might see a passenger who tripped falling in slow motion, because the forward momentum of the passenger + the speed of the craft must never exceed the speed of light, while the passenger would experience no effect of this time dilation, and would simply trip and fall.

Which brings me to my question...

Black holes, even supermassive black holes, are nowhere near the mass of the universe as a whole; so when the universe itself was much smaller and far more dense, then at a universal scale time itself would have progressed more "slowly" than it does today, and since the expansion of the universe is accelerating, doesn't it mean that time is also accelerating?

(Assuming, of course, there was some "outside" to observe this from).

This is not a duplicate of Does time slow down because the universe is expanding at an accelerating rate?.  That question and its answers seem more concerned with local, relativistic time (which is as malleable as space itself).  My question asks more about the effect across a universal scale as it relates to density of mass across that same universal scale.
The rate at which a clock ticks is a local property. There is no universal rate of time which could speed up, so the answer is no.

A clock stationary in the same frame of reference, and the same gravitational field as me will tick at the same rate, (one second every second) It is only clocks that are moving, or in different gravitational fields that that tick at a different rate. If I am falling into a black hole, and I stop the check the time, I wouldn't see my watch slow down.

So there is no standard clock rate. Clocks in the early universe (or at least things that depend on time, such as nuclear decay) ran at the same rate, in their local gravitational field (at one second per second)

If we were to observe a clock from the early universe, it would be receding at great velocity (and so it would be redshifted and time dilated) 

Now, the gravitational time dilation does not depend on the amount of mass directly, but on the intensity of the gravitational field. A galaxy has huge mass, but the only place in which the gravitational field is significant (from a GR perspective) is in the neighbourhood of Neutron stars and black holes. 

To get gravitational time dialtion you need an intense gravitational field. Just having a lot of mass is not enough. After the period of inflation, the universe was homogeneous, and there were no significant "lumps" to give a net gravitational field. As the gas in the universe collapsed into galaxies and stars, and eventually black holes regions of intense gravity formed, in which clocks would run slowly. 

However there is no general speeding up of clocks in the universe.