I'm sorry OP, this sounds hard.
It's normal to feel protective of your dc, and I suppose the one happy thing to take from this is that it seems you have built a warmer bond with your DS than you had with your father, for which you are to be congratulated.
IMO, the truth, as is so often the case, is that you are all a bit at fault.
A five year old should not be burping at the table, and I'm afraid I find nothing more irritating than dc who can't rise to answer a question with politeness. Shyness can be very real, but a couple of words acknowledgement should be manageable by age 5, even just a "fine thank you." Perhaps work on some easy responses for when he gets a bit tongue-tied. My grandmother used to say the cure for shyness is to think about the other person's feelings, not your own feelings of shyness, though I understand it can be hard to overcome. It will stand him in good stead if you support him with this, because shyness can come across as exasperating recalcitrance.
I think you were probably being a bit gentle with him - and from what you have said about your relationship with your father growing up, I totally sympathise that you have tried a different approach. But I think if he is wriggling at the table and burping, you need to crank up the firmness a little. We all have to adjust as parents when we aren't getting results, so don't feel crushed.
Your Dad is clearly a grumpy person generally, but having people in the house can be hard when you are not used to it - especially if they belch and don't answer, and then are not being corrected! And weirdly, at the back of it he may be genuinely concerned and trying to shock you into addressing it as a way of helping you. I know quite a few characters like that: they find it hard to do that sort of thing in a loving way, but there is concern at the back of it. But making your DS cry is your Dad getting it wrong too.
I'd stick it out and try to leave on a civil note. I know it's hurtful when people attack your DC, but I think you will feel better in the coming months if you don't leave under a cloud.