Mine is the same, he's six and getting better. Reading, writing and a boost in confidence/skill at drawing have been the turning points.
I don't fill up his time with activities. His screen time is limited, but he does get a lot. Like you it's a guarantee of peace! We try to encourage him to do different things on the tablet rather than just games or TV - Lightbox coding app is good and so is Dragonbox maths, they're not gimmicky, but actually quite entertaining.
He does moan about being expected to entertain himself, especially if he's had a lot of screen time and then suddenly has less. Sometimes I go and do stuff with him, but often I want him to entertain himself more and he's got much better the more chance he's had.
It helped a lot recently when we weeded out his toys. So go through his room in categories (1. books, 2. activity/puzzle books + art and craft stuff, 3. games and "sets" (science kit, jigsaw puzzle, rubix cube type things, board games etc), 4. "pretend" toys (toy food, tool set, farm, cars, etc), 5. "building" toys (lego, marble run, train set thing), 6. dressing up stuff and gadgets (torch, measuring tape etc) and then 7. other or misc. You could add an "outdoor toys" category if you have a garden, we don't.) Soft toys don't count because they aren't really for playing, they're more friends. But add to "pretend" if he does play games with them.
Ask what they want to keep rather than what they want to get rid of. Get him to pick out his favourite (one, or three or five) and then a few more he wants to keep. I found that DS is a lot more likely to actually play with his toys when they aren't swamped in things he's outgrown or never really liked to begin with. (I haven't finished all of the categories yet but will soon). Then I also give him ideas for things to do when he feels bored - you could read, you could write a story, paint, draw, colour, build a marble run, do competitions with your cars, play a board game with your soft toys, I taught him a couple of single player card games too, and now he will moan for a bit but then go off and actually do something. And DH has found that his fallback of "You can stay in your room until you clean it" now has zero effect because DS is perfectly happy to sit in there for hours playing with a bouncy ball, then a train track, then a book, then back to the ball again 