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Added in response to the comments: My somewhat flippant answer has gotten me in trouble;let me see if I can extricate myself.

I think that everyone agrees with finite choice: if we have a finite collection of non-empty sets$S_1,\ldots,S_n$, then we can surely choose an element $s_i$ from each $S_i$

Now suppose we have a sequence of non-empty sets $S_i$. Then, if you can imagine the set of natural numbers in its entirety (in my mind it is laid out as a set of stone, one following the other), then you can image passing step by step through the natural numbers, choosing an element from each $S_i$ in turn.

Although personally I don't have any particular objection to full choice, I can see why the consideration of a general infinite process (choosing an element from each of an infinite collection of sets) could seem much more unreasonable then choosing an element from each of a sequence of sets. Although this is not the direct logical relation of well-ordering with the axiom of choice (which involves well-ordering the union of the sets, not the indexing set), I nevertheless maintain that an infinite process seems more reasonable if you can imagine it being taken one step at a time, rather than if you imagine it being indexed by some amorphous, unstructured infinite set.

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It might be relevant that no-one has trouble well-ordering a countable set.