OP, I also feel your pain. Your original post just made me cringe on your behalf. If other people want to do things, and they want to gently encourage others to join them, that's fine; but repeatedly pushing when someone has declined, or trying to shame them into joining, is rude.
My extended family is musical and are somewhat into performing at family gatherings. I am NOT musical. If they ever try to pressure me into singing along, I shake my head, smile, and say "no thanks." If necessary, I repeat this firmly. If they ask too often, I get up and walk away from the center of activity. I try to always keep smiling, even if it's obviously forced, because as O'Dell said, it's hard for people to complain that you're being unpleasant if you keep smiling.
You said you've tried doing things in the kitchen to keep busy, and people keep seeking you out. If you end up having to walk away from the center of activity, is there, perhaps, an upstairs bathroom you could go to? At least until they give up and leave you alone. It would be pretty heinous to knock on a bathroom door and demand that someone come downstairs and sing with the group, when they don't know if the person might be ill or something. If I've said "no thanks" and people won't leave me alone, I'm not above hiding somewhere, frankly.
Another option is to seek out a relative who has a similar opinion, and arrange to go for a walk with them at the time of the performance. You'll end up having a nice visit with that particular relative, and "blame" is easier to deflect with someone else standing firm beside you.
To me, it sounds like the aunt is pressuring people to do one particular activity at the gatherings--sure, some people are singing, some playing instruments, some making up games for everyone, but the way I see it, it's all forcing the whole group to interact in a defined way. I get to know people better by having extended conversations in small groups, about whatever topics that come up naturally. If there was a very small amount of performing, maybe half an hour or so by people whose personality makes it enjoyable for them, and then we went on to another kind of interaction, I wouldn't mind the performing at all, and I would probably think it was pleasant to witness. But to me Aunt crosses a line when she insists that it be the only kind of interaction allowed, and insists that everyone participate in it actively whether they want to or not. It's a family holiday gathering, not a summer camp.