That's an interesting point. Looking at my own family, any pushiness seems to happen per-relationship, and per-context. For example, I see something my son likes, I remember liking that thing myself, and then I recommend some extension to that thing or say "you like that? Try this!" I mean, I'm not the stereotypical tiger parent but I'm sure I seem pushy at times.

The pushiness definitely happens in both directions though. I have heard enough "dad, you should"'s that sometimes my wife intervenes and tells our kids that, no, Dad is fine how he is. I like to try things my kids like, so they think I'm a kid sometimes too.

Speaking of screen time, I have one son to whom shared screen time is a genuinely motivating pleasure. He just beat Sonic Adventure DC after blowing all his precious screen time minutes on it for a few months. When we play a game together, he howls with laughter and loves griefing. Another son is indifferent about where the shared time happens, so long as he gets equal time with Dad (it's really the fairness and shared experience he cares about). I taught him a solitaire pyramid clone we played with Star Wars playing cards and called it Star Destroyer. He played it all day.

My daughter I'm lucky to bring down out of Earth orbit every once in a while; she is constantly dreaming and I admire that capacity. "Oh it's my turn to play in your office dad? Can we have a dance party with that one song...?" Neil Diamond's "America" for some reason...

The three of us play little narrative group RPGs together. I make them up; they are a little like those old game books that were a tiny step up in complexity from CYOA. The kids automatically project their life approach and personality into their characters. Occasionally I have a little GM trickster fun like transforming their favorite weapon into a burrito, but the tolerance for this varies greatly by kid. Sometimes if your favorite archetype is Hulk and you had a hard day at school, you just need time and space to be an epic Hulk.