> is it damaging if your caregiver uses phone a lot but remains responsive.

> but that is the question I find most interesting as someone who uses my phone to get away from my desk but keep work going so I can spend time with my kids while doing my job.

This is probably more relevant to people who are using their phones for passive entertainment than work. The difference is that your kids are more engaging than your work, so your attention will naturally be pulled toward your kids, whereas parents who are scrolling social media or playing games are going to be more drawn to their phones and are going to have punctuated and shallow interactions with their kids.

Personally, I don't think I've ever observed someone who was actively engaged with their phone who was also fully engaged with the people around them.