There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms.
However, there are also more mundane issues. I think that, whether or not it's ideal, a key point here is usability.
An answer like this may seem dubiously appropriate at philosophy.stackexchange, but I think it's an important part of the picture - and any philosophy of mathematics that doesn't take into account actual mathematical practice is fundamentally incomplete, in my opinion.
First, I think it's easier to recognize an implicit use of tameness assumptions (like "every set is measurable") than it is to recognize an implicit use of choice. (This is especially true because there are tameness assumptions which are secretly applications of choice, like "every vector space has a basis"). What this means is that adding AC as an axiom to ZF makes it substantially easier to recognize a natural-language proof (that is, a proof as written by human mathematicians, as opposed to a fully-formalized proof) as valid (= demonstrating that the principle in question is actually a consequence of our axioms).
That is, there's a distinction between improving proofs and improving results. Even if we accept that choice leads to undesirable consequences moreso than its negation (which I'm not sure I buy), this doesn't address the issue of whether "proving in ZFC" is an easier/more natural task than "proving in ZF" or "proving in ZF + [tameness property]."
For a concrete example of how theories involving "taming" negations of AC can be hard to use, consider the most common AC-alternative: determinacy and its variants. On the one hand determinacy does prove that every set of reals is measurable, has the Baire and perfect set properties, etc., although these proofs are nontrivial. On the other hand, the justification for determinacy is surprisingly fragile: "every game has a winning strategy" misses the fact that the set of moves is fundamentally important - determinacy for games on the countable ordinals is outright inconsistent with ZF!
So in order to actually get tameness out of determinacy we need to spend some work learning how to use determinacy (which is much harder than learning how to use choice, in my opinion); meanwhile, choice is a much more "global" axiom, and the naive justification for choice isn't actually misleading in contrast with that for determinacy.
Note that this is a different question from whether AC is true (whatever that means), but I think we can't ignore pragmatic concerns in studying mathematical practice.
Now coming back to the italicized third paragraph above, an important question at this point (especially for this site) is whether we can extract from this pragmatic concern an actual philosophical argument or observation.
I think in this case we actually can. When we reject choice on the grounds that it implies the existence of "pathological" objects, we're making the implicit assumption that mathematical objects are more fundamental than mathematical proofs, and I think this is unjustified.
Moreover, when we get down to it the objection to choice that you're making is also significantly pragmatic, at least until we've given further justification: e.g. from a Platonist perspective, why should pathological behavior imply nonexistence?
For what it's worth, there is a precise sense in which the issue of AC doesn't affect "concrete" mathematics.
Shoenfield's absoluteness theorem states that any Pi^1_2 statement true in M is true in N, whenever M and N are models of ZF with the same ordinals. This is a bit technical, but the key points are:
Basically every concrete mathematical principle is Pi^1_2 (or indeed much simpler).
Godel showed that for every model M of ZF there is another model L^M with the same ordinals such that L^M is a model of ZFC.
As a consequence (via the completeness theorem), every Pi^1_2 theorem of ZFC is a theorem of ZF, and so basically every concrete mathematical principle which is ZFC-provable is also ZF-provable. Similarly, the continuum hypothesis doesn't affect Pi^1_2 principles.
