Involvement v Independence

Yesterday was rainy, so I got a rare chance to come home early due to baseball practice being cancelled. I nice surprise. Spent some time rolling around the floor with my daughter (Megan -1o months old). She is getting so big. Crawls around faster than anything, and thinks she can walk like a two-year-old. She constantly wants me to hold her hands so she can walk around. It's fun for a while, until my back starts killing me from being hunched over. And after 10 minutes or so, honestly, I had enough. I'd much rather play on the floor with her toys. She is much more independent than my son though. He needed constant attention. But from what I hear, that is pretty much normal for the first child. My sister (a middle child) has already warned my daughter about the negatives of being a middle child (now that our third is on the way). She's nuts. But both my son and daughter are very happy children. We try to give them a lot of attention and be involved in their play. Which brings me to my discussion topic of the day:

The balance between attention/involvement and independence.
I know that kids need to learn how to play independently. As my son has gotten older, we've kind of encouraged him to "play by yourself for a while". This is tough since he had never had to do that much before. We always played with him. Maybe too much. But sometimes I think that is good early on. For example, when my wife and/or I take him to the park, we are climbing the playground equipment, chasing him and are very hands-on. He loves it. I think some parents would say "they need to learn how to play by themselves", which is another perspective I respect. But when other kids see us playing with our son, they often approach us and ask to join in or play with them to. Or even say things like "look what I can do". All as their parents sit on a bench somewhere, sometimes on the phone, or whatever. I feel bad that it seems they are craving attention. But maybe they need to be playing on their own. Sometimes I make my son do it, just to get a little balance. I don't know what the right answer is, but I know he loves the attention and us being involved in his play. However, there is the downside of him getting upset when we can't. Sometimes when we are making dinner, or paying bills, or feeding our daughter, we tell him he has to play by himself a while. Most of the time he's fine with it, but sometimes he says "But who's gonna play with me?". Sounds sad, but I think it's something we created. That's where finding that balance comes in I think. Anyway, something to think about.