Ok, I'll bite. I am not my children's playmate nor do I want to be. I do not judge people who do this. This is what makes sense for our family. While I am not a Continuum Concept person (I hated that book), I see some value in a modified system of benign neglect. I love to see the wacky things my kids come up with when left to their own devices (you should see what they can do with clothes hangers!). I should add that this is different when you have more than one child, however. My kids are each other's playmate. Another factor is personality. My oldest child is an introvert like me. I cannot tell if my youngest is introverted, but he does not crave playtime with other kids (so maybe yes?). He's very friendly with them and will ask them what their names are, etc, but he doesn't ask to go see other kids. So, there are lots of variables.
I always feel a bit guilty for saying this, but I don't enjoy playing most things. I enjoy speaking with my kids, doing art with them, taking them to fun places like the museum, reading to them and engaging them in those sorts of ways. But I do not want to play trucks or try to exist in their fantasy world. I will throw an imaginary rope to a child stuck in some imaginary predicament or I will talk to someone's imaginary friend or eat some imaginary blueberry sauce but I don't want to be *in* their game. Honestly, it would drive me nuts and I have other things I want to do.
Besides not wanting to play, I do worry that it takes away their self-motivation to entertain themselves. It's very important to me that they are able to solve their own boredom and stretch their brains to be creative. I'm not saying that other kids don't do that. I'm just saying that I believe the modified system of benign neglect encourages that, IMHO. Ds1 went through a brief but frustrating stage of wanting us to draw for him or tell him what to draw. That really bothered me. I want him to use his own imagination and not look to us to tell him how to perceive things. When he stopped that stage, he imagined and drew much more fanciful things than I could ever imagine.
I also don't see this as an issue of "my child is too sophisticated to play with age-mates". My oldest disliked age-mates for a long time, mostly because he was nervous around them. But he loved talking to adults and older children; he loved the discussion and the attention. But that never translated into him wanting me to be his playmate. When my second was born, they were inseparable from almost the start. When my youngest was able to sit and play, my oldest would play with him on the baby's level. When he became a mobile 1 year old, my oldest would seek him out to involve in his games; the youngest was basically recruited to be a play assistant. I have noticed that my oldest tries to engage other toddlers in this way. Even though they may have limited speech and be babylike, my oldest loves playing with them. So, while it's only anecdotal, my oldest seems to enjoy playing with those above and below his level of sophistication. With agemates, with whom he finally became comfortable and less timid around, he often becomes the leader by devising imaginary situations for them to deal with.
I think the desire for a parent to be a playmate must be rooted in many areas, such as birth order (maybe onlies are more like this?), personality, introversion vs extraversion, and family practices. It's not something we do, because I don't want to play a lot or be responsible for entertainment and because I don't think my kids really want me to be in whatever they're doing. If it's what works for someone else, then that's great. It's just not my style or theirs. We do have meaningful interaction in other ways, however.