Surely only a Platonist could believe that AC is true; we philosophically sophisticated people know that you can make whatever assumptions you want!

Yes, indeed!

And so naturally a theorem with a proof using AC is a weaker result than the same theorem with a proof that doesn't, since it holds under fewer sets of assumptions, and thus the latter is preferred.

Yes -- but it needs to be stressed that this doesn't distinguish AC from anything else! (Also, depending on the context, there may other criteria for selecting proofs besides the strength or weakness of their assumptions.)

If only people would talk about whether they prefer working in ZFC or ZF+not(C) (or plain ZF), or better yet what they like and don't like about each, rather than whether AC is "true" or how "skeptical" they are.