I’m not disputing that gamification is a problem, but kids have been getting angry at parents for taking their toys away since the dawn of time.
If a kid is deeply engaged with an activity, suddenly terminating it for arbitrary reason is a huge sign of disrespect from parents. Kids get as angry about this as adults, it's just that adults are trained to repress the anger. Consider: if you, as an adult, were immersed in a book or in a team sport, and I came up and ordered you to stop what you're doing right now and go do something else - wouldn't you feel at least a bit angry?
Really I think a lot of the “my kid got mad because X” comes down to setting proper expectations. Whenever we set expectations with her, and especially when she gets to participate in the setting, things go so much more smoothly. Not to say we’re perfect, far from it, but the contrast is striking.
My wife is inclined to give "bonus minutes". I'm inclined to give warnings, as I've found the effect the same. My wife will come out and say "It's been ten minutes, we're moving on to X" and toddler will struggle so she'll say "two bonus minutes". I'll remind him after eight minutes that "he's got two minutes left". He'll ask for bonus minutes, but when I say no there isn't much of a struggle.
Although I know I’m going to need to teach him how to live in our screen-filled world, I’m not sure when I’m going to feel like he’s mature enough to face the intensely addictive properties of the thing. Frankly, my personal experience is that I’m not ready to appropriately handle their addictive properties for myself even now, so...
Our working theory is that by not making screen time precious, we’ve made screens just another toy/tool/way to accomplish something, rather than something to be sought out for their own sake.