Short answer - you don’t.
Maybe Mum just wants to spend some quality time with her son, maybe she has other reasons, but whatever they are, it’s not really something you need to question. All this play date/ clubs/activities stuff that some young parents get caught up in is neither necessary nor compulsory, and can end up being pretty expensive. And maybe, just maybe, she’s worried that you seem to favour the older boy over the younger child.
I was a single parent, and I used to love the summer holidays with my kids - we had long earnest chats over meals, danced round the kitchen to silly songs, cooked together, played French cricket, swingball and boules in the garden, had long evenings with board games and canasta, and sat up late with a telescope stargazing. They got all the socialising they needed during termtime, and the holiday was all about family time and building happy memories. They occasionally went swimming with friends in a local millpond or kicked a football around, but that was all. They now look back on those holidays with pleasure.
If any of them had been asked to spend 10 days on their own with either grandparent, they would have flat out refused. Have you considered that a seven year old child might express a reluctance to spend that much time with an ageing couple in an unfamiliar place and having to be on best behaviour to please Granny?
Being asked for an honest opinion
