Many people spend a lot of time worrying about being polite. If by polite, one means being civil, I have no problem with that. I much prefer civil discourse to a shouting match. If by polite, one means saying "please" and "thank you" I have no problem with that. If by polite, one means saying "pardon me" when bumping into someone accidentally, I have no problem with that, either. Politeness is a set of conventions designed to prevent people from becoming upset with one another unnecessarily, which is all to the good. Being polite is a very human way to signal our good intentions.
Where I have a problem with politeness is the supposed convention of saying "excuse me" after some natural bodily function. There can be a large difference between an exaggerated belch, done for effect, and a simple burp. If someone is attending a solemn ceremony and belches unnecessarily loudly, does saying "excuse me" paper over what they just did? Does it make it O.K. to do such things? Intentionally disrupting a solemn event is usually seen as rude and signals a disrespect for the occasion. Mere words strike me as ineffectual when seeking atonement and irrelevant to the situation.
On the other hand, if someone simply burps in a relatively discrete way, I see no reason for apologizing or begging for forgiveness. Burping and flatulence constitute a natural consequence of the way our bodies work, so allowing those functions to occur isn't being impolite. It's just being human.
When a group of boys get together, it's almost inevitable that the conversation will come around to passing gas from the rear end of the digestive system - farting. In the silly world of boys (and men), belches and farts become something of pride, and the more odorous (and odious) and louder, the better. For some mysterious reasons having to do with our societal conventions, when girls fart, it's supposed to be quiet, discrete, and characterized by a smell akin to a rose garden - all untrue, of course. As most any married man can testify, most wives feel much less inhibited by such conventions with their families and cut 'em off just like a sailor, and then giggle about it. Saying "excuse me" after such an event strikes me as superfluous and even hypocritical. I'm on a hopeless one-man quest to eradicate the necessity to excuse oneself for acts made inevitable by living. And if they're obviously done with no effort made to be discrete, I'm even less inclined to accept "excuse me" as an excuse.
I don't advocate loud, noxious gaseous explosions in solemn situations. I know that the sound of belches and farts can be controlled to a certain extent, and I agree fully that it would be inappropriate to exaggerate such acts in some situations. Sometimes, however, the intensity is unanticipated by the person needing to expel that gas (from either end). It might well burst forth at a bad time. Asking for forgiveness afterward would be a natural consequence of the embarrassment associated with unintentionally calling attention to oneself by such a loud (and/or) malodorous act. Saying "excuse me" under such circumstances makes sense to me. The act amounts to an inadvertent "social mistake" and if one truly feels ashamed or embarrassed by that act, then it seems perfectly logical to ask for forgiveness.
But I think such forgiveness might be more difficult to grant if the deed were done with evident "malice" aforethought. No simple "excuse me" can atone for doing something intentionally disrespectful. If the same acts are done intentionally but in a situation where no real embarrassment is involved (e.g., between spouses), then the "excuse me" is again completely unnecessary. There are times and situations permitting loud belches and farts, and learning when to do them and when not to do them is part of becoming socialized. It seems obvious to me that learning when to say "excuse me" is also part of that socialization - it's just unnecessary in many situations.
I note in conclusion that in some societies, loud belches after a good meal aren't indicators of disrespect at all, but rather signal the host that the guest had a satisfying meal. So what is polite is simply a social convention and can vary from one society to another. So long as everyone understands the intent, then it works.