I like the library idea, too, and for children the library fees often are zero or at least reduced, so that it is an inexpensive task. Some libraries also lend out board games or other games.
I am not sure if it works for your son due to the screen aversion, but many libraries have e-books and magazines on apps like Libby or Borrowbox, which saves you going to the library for now.
The idea of the activity tombola sounds so good, I could use it for my own household chores 😀
I think many ND people struggle with starting tasks, so that the "try this for x minutes" approach can be helpful.
If you think he likes the challenges of interesting questions or being genuinely helpful or physically moving something (like the bricks and finding out about how something measures in Lego bricks), maybe instead of chores the helping could be framed as helping his mother because of the pregancy she needs more help for some weeks, or maybe he would like to help other neighbours with simple things (an errand for someone who is not mobile, or watering the plants for someone).
The difference between simple chores and help is that there is a reason.
Some people do better when they know the reason and are pleased to solve a problem, as some PP hinted at as well.
I do have a warning a about Minecraft, even if it is something creative, like world-building and the likes, because some ND people get addicted to stuff quicker than NT. They can react more strongly to the rewards in games (do not mean things like online chess, which sounds helpful in this case, but mainstream computer games).
Maybe the best is some combination of learning and help, such as a scavenger hunt in the neighbourhood - find 5 different insects / birds /plants and tell me about them / document them, something like that, in line with his academic level, plus doing helpful / useful things that also help you in your situation.
Can he make his own timetable and follow it? Try a task to make his plan for the coming days?