I note you haven't been able to say what the awful connotations are, OP, or in what sense it's rude or crass.
Try to be less judgemental about the words people use. It's often not meant to offend.
For me, belly is a perfectly fine way to describe the part of the abdomen that contains your guts, or, in case that word's offensive, your viscera or your digestive tract... or maybe you prefer intestines? Bowels, maybe???
For instance, in most cases it'd be more accurate to say "I have a belly ache" than to say "I have a stomach ache" - as those pains are rarely coming from the actual stomach.
A friend of mine was once turned away from A&E here in France because as he was saying "my stomach hurts" he was pointing to his belly. He should maybe have said "ventre". Anyway, they said to him "that's not your stomach" and pretty much something along the lines of "come back when you've learnt some anatomy". It was NOT a busy A&E.
People can be so dismissive of others. Honestly, it's a word like any other.