I don't think it's unreasonable to feel nervous/scared about this. That's a perfectly normal reaction. I'm guessing that your family also know you don't do well with this sort of thing.
My older sister, who is on the autistic spectrum, is the same. When my dad was very ill (and ultimately did die), she was hopeless. Didn't/wouldn't/couldn't go and visit him. Left all his care to my mum and I.
I know it wasn't her fault, but I was really angry about it for a long time. I was struggling with my own stuff at the same time, and just felt well and truly dumped on. And I know it'll be the same when our mum needs extra support/dies. I've accepted it, but it's done our relationship no favours. I know we've got some hard conversations ahead in the years to come.
I get that it's hard for you. But it's also hard for everyone else. If you really CAN'T do the "sitting at the bedside" thing, what CAN you do instead? Can you sort out the behind the scenes stuff? Things like making sure your mum has a meal in the freezer for when she gets home. Or calling round to get quotes for any house adaptations that might be needed? Making sure the hospital visitors have cash for parking machines? Etc etc. Please don't just bow out completely.
My sister came out with all sorts of corkers, like "I'm asserting a boundary I've put for myself, and not coming." All well and good, but my mum and I never got a chance to "assert a boundary", because there was Shit To Do.