Why is Keynes attacking the (neo)classical theory of interest rates on the grounds that it is indeterminate? I do understand that he argues that in order to draw the loanable funds supply curve, you have to already know the volume of income, i.e. production, and therefore the interest rate in advance.
But why couldn't you just test for coherence in the following way?
Assume a level of income. Draw the corresponding loanable funds supply curve. Calculate the interest rate at the intersection of the supply and the demand curve, i.e. the interest rate for which the capital market clears. Determine if the income level corresponding to this interest rate is indeed the income level we assumed in the first place. If it is, this interest rate is an/the equilibrium rate, if it's not, it's not.
Is there anything wrong with this argument?
You can use some some initial $Y_0$ to find an interest rate, but this is precisely the indeterminacy problem.  We would like to be able to derive the equilibrium interest rate without assuming some initial values.
Furthermore, note the indeterminacy problem is actually present even in canonical Keynesian models (See Hansen, 1951) and Neo-Keynesian models (see Beyer &amp; Farmer 2004).
However, the Keynes critique is now a bit outdated. Keynes was writing in a past before macroeconomic was even given proper formal treatment (by modern standards), and at a time when macro had no micro-foundationds. Nowadays you can solve the indeterminancy problem in macroeconomic model of either variety by assuming central bank follows some interest rate setting rule, such as Taylor rule or making some specific assumptions about model's micro-fundamentals (see Romer's Advanced Macroeconomcis for discussion of this). Although as the Beyer &amp; Farmer point out in their paper even central bank interest setting rules won't solve this issue always so the problem is somewhat open.