Setting aside the question of cursive and handwriting, I'd say "screen time" is more a concern in general overall, inside or outside the classroom. I may sound like some old fart, but I don't think so. It applies to myself as much as anyone, let alone kids growing up with constant access to the tech.
I make a point to be more observant in life, for instance as I'm out walking around, or just sitting on a bench in a park, etc. It has to do with being a part of the world, feeling connected with it. Being present in the moment. And I look around and see people walking by the beautiful lake with their faces down towards the screens on the phone, headphones on, and totally not present in the moment whatsoever.
This sort of thing I fully believe creates anxiety for us. It's bad enough when we are lost inside our heads, inside of "thought world" as I call it, and not in the real world surrounding us, but then add we're walking with our heads down facing a lit screen, reading texts, and just walking around periferrially just enough to maybe avoid running into other screen-walkers, I'll now term them. This is a **** way to live.
So as a child growing up, developmentally, I cannot image this is psychologically healthy for them. I can't believe it is socially healthy either. And if these are out of whack, then so too will their bodies eventually be effected in negative health down the road. I believe in a whole body mind system balance. Too much tech can easily put things out of whack.
Granted it, we have to live with tech. It is part of our world now, I'm even on a screen typing right now, but it has to be countereffected by other, non-screen related activities - especially outdoors! I'd say they need to take off the blinders and be present in the moment. Be part of themselves as part of the world around them.