I feel like I've lost all sense of normal with this. DS1 age 10 is so needy for attention at weekends or during holidays, or even after school sometimes. We have recently cut his screen time right down - it had just slipped and slipped until he was on it for the entire day which was ridiculous. So first I cut it down to 6 hours max (as in less than half awake time) but it was still too much so I've cut it in half again and now he has 3 hours. He mainly uses this to play Fortnite with school friends or play single player games on the playstation/switch or watch YouTube on his phone. (Yes - he has a smartphone - but this is fairly typical among his school friends, we're not in the UK. It's got controls on it and we remove it at night.)
For the first (half term) week we cut it down to 3h I mentally allocated myself 3 hours of time to occupy him myself too, to kind of help him wean down from the screens and kick start him into doing stuff. Made the mistake of mentioning this to him and he just ended up badgering me for the entire week "What are we doing together?" "You've only spent half an hour, I've got 2.5 hours left" I ended up exhausted and had no time to get anything done 
When he is meant to be occupying himself he can think of "literally nothing" to do, all he does is flop around and complain about being bored, or hover around near people generally being in your personal space but not actually doing anything or making demands to "Doooo something together". The one thing he will do when he gets fed up enough is read, but he doesn't go off by himself to read, he stands behind me reading the jokes from a comic type book in a monotone because it's like he can't fathom the idea of simply enjoying something for himself. I feel like a massive arse and a horrible mother for essentially wanting him to shut up and leave me alone for a bit. But I'm sure by 10 I'm not meant to be providing constant entertainment or activities for him? Am I? We do stuff together but not all day every day. I do wonder if I should set up crafts or something, but I would have thought he'd be a bit old for things like that now. Should I be stuffing his days full of planned activities? I've tried to get him to sign up for clubs and things like that but am just met with total lack of interest. He keeps telling me he only likes doing new things, but doesn't actually want to try anything unfamiliar
I tried making a rule that he was not allowed to ask to "do something" but he was allowed to make a suggestion e.g. "Do you want to play monopoly?" that was a bit better, but he still wouldn't go off on his own and do stuff. The baby is better at playing alone than he is (It's not related to sibling jealousy BTW - he was like this long before any sniff of a baby.) In fact the baby slightly improves things as now he has someone to play with, but we do sometimes end up telling him off because he gets too hyper and silly around him.
Do you entertain your 10yos? What kinds of non-screen things do they go off and do on their own?