I'm not going to get into all the ins and outs of this, but I'd just like to point out the very massive difference between having people pop by in your own place and having to go somewhere else in the early days. Or going for a walk outside, at your convenience, when you know the baby has just been fed/slept and is in a happy place.
It's MASSIVE if it's your first child and you're sleep deprived, overwhelmed and just not used to your tiny baby yet. Just packing up all the stuff you possibly might need to go away for a few hours is stressful and exhausting. Committing to a timeperiod of when to be somewhere when you don't know if your baby will be hungry, wet, tired, overtired or sleeping is even more stressful. Newborns don't have any routines yet, so for all you know the baby could be needing to feed or sleep the entire time she's there! It's just such a massive stressful hassle altogheter, I'm not surprised she says no.
And add to that the fact that there'll be 8-10 people there all wanting to hold the baby, talk to the baby, talk about birth, discuss your parenting techniques, blahblahblah, while you're trying to make sure baby is ok, maybe trying to breastfeed or trying to get him/her to sleep.. Or that she'll be reliant on you to go home, so might be facing another possible battle on that front because you (or everyone else) thinks she should stay longer.
I'd never do it again. We went away to ILs house 5 weeks after DS was born and it was hell on earth. I had a massive breakdown and it was the most stressful day ever.
Disclaimer: I'm not saying it will be like this for everyone, but it could possibly be so for her. And even if it isn't, it is very likely imagining the day goes like that in her head.
So give her some space and time. And if it's so important for your family/parents to see the baby (which I get!), then FFS they need to get over their awkwardness and go see the baby. It is just not fair to make her go to such lengths when they're perfectly welcome to go visit.
Oh and DH (who I live with) didn't go anywhere with DS on his own in the beginning. It just wasn't feasible. Of course he'd have him on his own at home, but mostly for the length of a shower / visit to the toilet... I went to training at 6 weeks and that was probably the first time he had him at home by himself - for all of 40 minutes!