I have two teen boys -- nearly 14 and a freshly minted adult. We had some rough patches, but generally they remained engaged with their family and outside responsibilities. I would like to gently suggest that you are approaching this the wrong way.
Quit seeing video games and screens as the enemy, instead consider them as the tools they are. Otherwise, you are fighting a losing battle. The first task seems to be that you need to reconnect with your children. You may still be operating in the outdated parenting dynamic of raising younger children. As teenagers, they are young adults and things go so much more smoothly once the parent adjusts to that dynamic. It was very hard for me with my eldest -- I had to send him to camp for two weeks in order for me to reset and see him as a 14 year old young man instead of as a child.
Next, embrace your children's interests. Often, our kids already have a spark lit inside of them but we don't see it because it's not our spark or we don't think the spark is good enough for our child. One child loves video games? Put in a request for him to choose a game he thinks you both could enjoy, then set up a weekly "date" to play together. Hell, you could even supply a few special treats or forbidden snacks (woo, soda!) to make it more fun. The screen won't be the enemy anymore, it will be the jumping off point for a shared interest. After a few weeks of this, don't be surprised if he becomes more open to other activities.
Another idea is to combine the love of screens with something you enjoy. Geocaching is the first thing I can think of. You use an app to find hidden "treasure." There are physical caches you can find along beautiful hiking trails, urban caches you can pick up cruising around the neighborhood, and various cultural caches that take you to hidden gems in your area that have natural or cultural significance.
Finally, we quit ultimatums around age 13. We had honest talks about how they were in the new phase where it was time to start learning how to be adults by practicing. Unless there was threat to life or limb, we delivered few rules without first discussing it with the kids. Some parents are against this, perhaps afraid of the power dynamic shifting. It worked well for us, though. Instead of being seen as the enemy, as so many of their peers see their parents, ours treat us as more of collaborators and mentors. Sure, it means we get a few more uncomfortable conversations that we would rather avoid, because the kids share (almost) everything, but it's well worth it.
Also, don't be hurt when they need to withdraw. This is a tough age, they need that alone time and even that mindless entertainment time. Keep in mind that previous generations needed this too, it's just their outlets were different and they achieved this my poking around in the woods, putting together puzzles, listening to music while laying on their bed staring at the ceiling for hours, or various other mindless tasks. Think of ol' Tom Sawyer staring out that schoolhouse window and daydreaming for hours on end. You can bet he would have been staring at his phone and doing that today.