coest--I had to delete the link to the product you're considering because it's a fundraising campaign.
But, to your point, I think if you just remove the screens from them and put them where the kids can't get at them, they'll adjust. I have had to do this when our family hit crisis mode and the kids had WAY more screen time (as a means of "electronic babysitting") than I was comfortable with or was good for them (I happen to have one that gets aggressive with too much exposure to a screen).
So for a full week or two, they will PUSH about having screens, but if you're devoted to whatever it is you want for your family in terms of screen exposure (none? some? scheduled?) then they will eventually see that you feel strongly enough about it to remain devoted to your decision and they will stop asking.
Certainly be ready to give them alternatives. My kids have a LOT of stuff (way too much even after I went through purging). Books, crayons, toys, Magnatiles... you name it. They certainly don't lack for things to do. And in my hasty annoyance, I have been known to remind them that if they don't want to touch any of those things, I could add them to the purge pile.
It's a lot of moaning and groaning. You need to figure out what you want their life to look like and just really stick to it. That doesn't mean you need to be nasty or mean about it, it just means you need to do what you think is right no matter how much the kids try to bully you into getting their way... much like everything else they want.