I have a 7 yr old. Her kindergarten required either piano or violin, and we opted for piano. At that time, we implemented a daily '5 minutes of piano' mini-lesson where I'd help her with her current lesson/song. We're doing homeschool now, but have kept up the '5 minutes of piano'. My daughter doesn't really like it that much, tbh, but I've made her stick with it because: a) she's actually quite talented -- great pitch/rhythm, and can already play by ear, for example, and I'd like to get her to a point where she can really enjoy that talent, and b) as a general lesson in perseverance/grit/skill mastery.
But I really struggle with the notion that the above is simply a rationalization and that I'm really trying make up for my own failure[1] to achieve my own musical goals vicariously through my daughter. And I've also struggled not to force it too much. I've bent over backwards to make it fun -- I make up games, do a variety of musical/rhythmic activities like drumming, dancing, or even playing pattycake, or do things like watching Fantasia or even listening to pop songs. And I've had to put the brakes on thoughts like "we're going to polish the crap out of this song so we can impress the relatives" and focus more on the overall process: is she having fun? On the other hand, I'm trying to not reward a "this requires some effort so I'm going to quit" type of mindset.
It's definitely been a tough balancing act.
[1] How I used to perceive it -- I no longer feel that I was a "failure" so much as I chose a suboptimal path.