It's not just oscillation but change in feedback up to that point as well, potentially effects output. Remember that in the feedback loop (the pair of resistors forming the voltage divider from output to inverting input) this is how the opamp "knows" what it's doing (not really, but for our purposes it's true enough) and corrects itself. The ideal is infinitely short feedback like soldering resistors direct to the opamp pins and to ground, but for practical purposes we like to lay it out nicely on a board but still short.
Use of other opamps with inherantly higher current tends to mean more work to keep them stable. Tangent has written
a bit on his website. As for what opamp you want, that's a whole 'nuther can of worms since many have their own sonic signature, particularly directly driving loads. Basically, to achieve more current is the starting point to move away from the basic CMOY and onto more elaborate amps. Truth be told, some sources already have as much current capability as the opamps being used in CMOYs. Then again I don't know how loud you need it, certainly some experimenation is in order if you had multiple opamps and socket it.
I doubt you will ever need a gain of 11, except if you had a passive crossfeed or other kind of passive circuit in front of the amp like an input selector.