I'm not sure an explanation would help an already fraught situation, unless it was something quite understandable - child is having night terrors, is going through a very clingy phase, etc - those I hope a sensitive grandparent would understand.
But every time I see grumbles about MILs I think of my poor mum who never intended or wanted to fall out with her DIL but somehow it happened.
My mums own mother died young so she never lived to be a grandparent or even see her children get married, so for my mum, who has also survived cancer, being a grandparent has been very precious, having this experience she was never able to share with her own mother. I think that's why she made sure we were close to our other granny, even though she struggled with her herself, and why I want her to be the kind of grandparent she'd like to be.
I don't think having GCs overnight is 'playing at mummy' and yes of course they don't do much while they're asleep - but presumably part of the fun is putting them to bed, bedtime stories and having breakfast with them the next morning. My granny had special china she got out just for us, and cooked our favourite dinners - we loved staying with her.
Having said that, though, if a parent has genuine concerns about safeguarding or whether a GP is capable of coping with a child, that's a different matter. My FIL is not in great health and I'd never leave a child alone with him - he'd never pick up a child for a cuddle or change a nappy for instance, and it isn't expected of him. If those boundaries are clear, you know where you stand. (I don't think he's yearning to look after DD alone, though he's fine to be around her when MIL is there too, and she's great with kids so have no worries there).