Another unschooling wannabe here. :)
I find that my 5yo daughter is mostly able to set reasonable limits on her own screen time. And if we do think she's getting too much, we can say something like "hey you've been watching/playing that a long time, how about you do something else for awhile?" and she'll say "okay" most of the time.
But my 13yo son has NEVER been able to set healthy limits for himself. We have tried lots of ways to give him more control, but it's been fruitless. However -- he has ADHD and Asperger's. This means that his 'executive function' does not work properly, does not engage like it should. So he has very little self-control, very limited skills on making good decisions. If he has an idea to do something, he does it, with no thought about consequences etc. This is not something that can be 'punished' out of him, it's like a learning disability. It can be 'trained' into him, but only in the very long term.
Our most recent experiment had him fully on board with WANTING to set his own limits. We decided to try it for a week. The first day, he did great! The second day, not so great. And by the end of the week he was a grumpy zombie, staying glued to the computer as close to 24/7 as he could, barely remembering to eat. Even though he *wanted* to do other things, he just couldn't manage himself.
So the pragmatic real fact is that he *needs* external guidance. He needs it in order to build up (through habit and practice) his own internal framework so that later in life he *can* have independence. Some kids just don't have internal structure to manage their own time.
Our current structure is in 'blocks'. He has a block of school work, a block of free time (where he can play video games), and a block of what I call 'project time' -- where he can do just about anything he chooses *except* video games, social media, etc. I have a list of suggestions if he needs ideas -- read a book, play with lego, watch a documentary/educational movie, do some composing/recording, do some computer programming, crafts, work on the puzzle, do some stop-motion film-making. All things that he's interested in and wants to do, but will *forget* to do when he's wrapped up in video games.
By alternating these throughout the day, rather than having all the free time at the end AFTER his school work and chores are done, he's been doing much better with applying himself. And because he can see that he DOES have free time scheduled throughout the day, he's been quite good about getting off when time's up. He feels better, he's in a better mood, he's happier. Even though he just plays for an hour then has to get off, having the structure puts him in a better place emotionally, so he resists less than if he'd been allowed to play for 3 hours straight while unrestricted! External structure is like a *comfort* for him. It gives his security and confidence. Without it, he feels lost and aimless.
So for kids with any kind of issue with self-regulation, an externally imposed structure can be *freeing*. It sounds contrary to unschooling principles, I know, but I believe that it's the better option for certain kinds of kids. Within the context of this external structure, my son has a lot of freedom to choose his own activities, and I follow his lead on academic choices as well, so I still try to follow the *meaning* of unschooling, the heart of it, if not the letter of it.
My husband (his stepdad) put it an interesting way recently that has really helped my attitude, too. He said, we don't want to *set him up to fail*. We have recognized and identified what his weaknesses are, so it's our job to remove those temptations, remove his obstacles, so that we can *set him up to succeed.* For instance, he has an iPod and when left unsupervised, will text and play games on it 24/7. When he's supposed to be doing something else. In the middle of the night. At the dinner table. Obsessively and addictedly. It distracts him from everything. So he's only allowed to have it in his 'free time' blocks. I would *like* to let him have it all the time and let him manage his own use -- I don't have a problem with taking a 5 minute break from your work to text a bit, then put it away and go back to work. That's perfectly normal IMO. However, *AT THIS POINT IN HIS LIFE* he's not able to do that. Giving him free access to his iPod all day long is setting him up to fail, because he WILL give in to the temptation and play with it all day. Treating it like that rather than as a 'punishment' for misbehaving has really made a difference in how POSITIVE we are all feeling about the situation. That's not lost on my son either.
Anyway, from the sounds of it, though, you're still looking for your son's hobbies and interests whereas I already know mine. Perhaps he needs some steering. You don't just randomly realize you like something until you've had a chance to try it -- or even to HEAR about it. ;) I know things are busy and hard, and maybe it's something that can wait until your daycare is closed (put up with the status quo until then). But I think you'll need to become more pro-active with your son and try different activities with him. Not just 'games' but all kinds of things. Maybe even enroll in some outside activities, courses, etc. Give him some opportunities to discover what he likes rather than just hoping he'll self-select something on his own. Maybe when the time is right, do a whole week electronics-free. This is HARD, I know! And the first day or two would likely involve lots of screaming and gnashing of teeth heh... but over the course of a week, you're FORCED to find other things to do.
Heck, even just making chores part of a daily routine can help. I only recently (last fall) instituted a chore list for my kids to do each day. I set a 'chore time' of 15 minutes at a time where we each do our assigned tasks. I hated the idea of it -- it's 'imposing my will' -- but THEY LOVE IT! My 13yo, who never wanted to do anything chore-like when I'd just ask, loves doing it this way. Does his tasks every single day with a smile on his face, and just that simple change made a spike in his positive attitude overall! They do feel better when they feel productive, useful, etc.
It's been hard for me, since I resist schedule and routine myself. And I hate doing chores. But it's for the good of my kids so I'm really trying to do better myself. I've definitely noticed that MY attitude and approach to the day rubs off on my kids.