The notion of countable and not countable does have some nuance. In the models that every real is definable that model, from within, believes it is uncountable since there is no bijection with a countable set from within that model. However, from the outside that model is countable.
The point is there is a lot of nuance and no matter what one believes regarding choice there will be unintuitive results. For instance, when the axiom of choice fails you can have an infinite set with not countably infinite subset. The reals can be a countable union of countable sets. There is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.
The point is there is a lot of nuance and no matter what one believes regarding choice there will be unintuitive results. For instance, when the axiom of choice fails you can have an infinite set with not countably infinite subset. The reals can be a countable union of countable sets. There is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.