One plausible reason for downvotes is that more than half of your question is a complaint about the axiom of choice that has no bearing on the question your post is asking. If you have questions about choice, they belong in their own question.

One plausible reason for downvotes is that a significant fraction of your question is a complaint about the axiom of choice that has no bearing on the question your post is asking.

Furthermore, your title sounds like you're seeking physical intuition about Lebesgue integration, but the opening paragraph speaks on a rather different topic, that of empirical testing the choice of technical mathematical details.

Your question would probably be better received if you picked a topic (ideally one you have a question about, rather than one you want to comment on) and stuck to it without drifting onto other topics.