It seems to come down to this:
If the GP are suitable carers for the child overnight (as determined by the parents, not the GP) by what age can a GP expect to have the GC on an overnight stay?
Some people seem to feel it ought to have happened by 3 years, others feel that at 3 years a child is still quite young and that it isn't unreasonable for parents to delay such sleepovers until the child is older. This may be because of practical or emotional readiness of the child for overnight stays, or it might be because the parent isn't ready, or simply sees it as unnecessary.
This seems to be connected to the experience of the poster growing up. Some stayed at grandparents from a young age, others from an older age and some not at all. People with all of these experiences still managed to develop close relationships with their grandparents.
There seems to be a double standard in the arguments in favour of the GP POV, where by a parent's desire to keep their child close to them at night is seen as selfish and precious whilst a grandparent's desire to have a grandchild overnight is seen as natural and to be encouraged. You can't have it both ways, either where the child spends overnight is emotionally important to those caring for the child and the child itself, or it isn't (in which case, why make a fuss about having the child overnight at all).
People are different and they parent differently. Not necessarily better or worse. My DN (my parent's first GC) has been staying over occasionally at theirs since he was 7/8months old and stopped BF. Whilst I respect my Dsis's decision, and see that it works for them, I think this would be too young for me to leave my DC overnight. Partly because I want to BF for a longer time if things work out, partly because I can't (yet) imagine wanting a break from the DC I spent 6 years TTC and partly because I have a different relationship with our parents. For example, before either of us had children, my Dsis would go around to DPs on Christmas day, DH and I have christmas day to ourselves and see boxing day or the day after for visiting relatives. I know its a small difference, but I guess its indictative of DH and I valuing our space as a little family, whereas my Dsis spends much more time with our DPs (she lives a bit closer as well). My Dsis and DBil also have lots of out of the home interests which mean that babysitting is very useful to them, whereas, my DH and I are more homebirds, and the kind of evenings out we do occasionally have, are ones suitable for a babe in arms.
My DM recognises this difference between me and my Dsis. She has let me know that she is always willing to lend a hand and is available both for short babysitting duties and overnight, and that she will respect our parenting style. (For example, when my Dnephew stays with them, she is tempted to have him stay up late so she can spend more time with him, but puts him to bed at his normal routined time as his parents wish). She's also made it clear that she's had her time as a mum (with 5 of us!) and doesn't want to impose either. I've let her know that I don't think I'll feel ready for overnight stays as early as my sister, but I appreciate her offer.
I do have some concerns about the suitablity of my DF to babysit, because he could be verbally abusive to me growing up. I realise he's unlikely to be this way with a small child, but its something I'd have to be very confident about before agreeing to overnight stays with him as the main carer.
My DMil is lovely, but lives much further away, and I don't really know about her parenting/grandparenting style as this will be her first GC. I'd imagine solo visits to her will be a long way off, but we'll try to keep her as involved as possible when we visit as a family.
Obviously there is another debate going on about the parent's judgement of how suitable the GP are to look after their child unsupervised. Its nice to know that some people have such great relationships with their family that they can't imagine not trusting them. Not everyone does. There's a difference between GPs spoiling kids by letting them stay up an hour or two later, or giving them some extra sweeties occasionally, and GPs having incompatible views about discipline, smoking, safety, allergies, criticism etc and who will ignore the parents wishes. Its an issue of trust.
In the recent past on Mumsnet we've had DD/DDils posting about being unhappy with GPs who don't take their children's anaphalactic allergies seriously, horders whose homes are unsafe but who want the GC there unsupervised, MIL who won't even speak to the mother of their GC on the day they come home from hospital and GP who see fit to take the child to a hairdresser without consulting the parents to have its babycurls cut off - and that's just the threads I've read recently and I certainly don't read all the threads. Some GP can't be trusted by the parents.
Others can, but the parents/children just aren't ready for extended visits.
YANBU to want your GC to stay. YABU not to respect that for whatever reason your Dil isn't ready for that yet. Concentrate on being someone she can trust by respecting her boundaries.