Others have already given great feedback on alternatives, but this hasn't yet been addressed:
I am trying so hard to limit time, but if I do so with my son, he will sit and do nothing - he can't think of anything else to do. I keep fighting this, trying to limit media time, but it's really really hard when they don't seem to want to do anything else on their own.
This sounds like classic manipulation. My DD10 has been doing this for years, where she asks us to do something she's perfectly capable of doing for herself, and when we refuse, she refuses as well. She does this because she's had a lot of success with this tactic... DW is a total pushover, and I still get drawn in where natural consequences aren't really an option. For example, we might do a living room camp-out, she refuses to get her own blankets, and just flops down without any to go to sleep. I can't let her freeze all night.
So while I obviously can't give you a foolproof method for dealing with this, I have managed to have some progress, so I can suggest:
1) As I mentioned, natural consequences. If he refuses to take up any alternative activities, fine. He can sit there bored and do nothing. If he tries punishing you in turn by complaining, whining, passive-aggressiveness, etc. don't engage. Let him know that his decision to do nothing was his choice, not yours.
2) Firm limits.
3) Use bonus screen time as a reward for other behaviors you've been working on... performing chores, getting ready for school on time, homework without complain, helping out, going outside to play, whatever.
