the not wanting them to depend on me for happiness is pretty self explanatory. same reason we don't use praise to control their behavior. just thank you if they do something helpful. or sharing our feelings if they upset us. i hope they will really get to know themselves without constantly looking back over their shoulder for me to be clapping/smiling. they deserve to feel intrinsically good about the things they do. not looking for outside motivation constantly. so the not playing fits into this. i know a mother who is the exact opposite of me and her children are incapable of sharing. they've had every single toy related conflict mediated by a "professional" and never learned for themselves WHY sharing is important. now she cant even leave her kids at their friends because they melt down with out a mediator navigating their conflicts with them.
i used praise with my first and she definitely suffers because of it. She will only do things when she thinks someone is watching and there to tell her what great work she did. Its very sad. My son. hah. He will grab a basket ball go down to the 10 ft tall hoop and at two years old throw the ball 25x without ever looking back to see what I think. And he will exclaim, "Happy" randomly when he's doing thing. It's just so refreshing to see how he doesn't need my approval to feel good about what he's doing. And thats really what "playing" with kids is. Telling them they are "doing a good job" through your actions and thats cryptonite for independence and intrinsic motivation.
Respectfully, I don't see how "playing with your kids" means they will depend on you for happiness. And I totally disagree that "playing with kids is telling them they're "doing a good job" through your actions". If that's how you feel when you play with your kids, then maybe you're doing it wrong...?
DH and I played tons with DS when he was little. Of course, DH and I also enjoyed board games, crafts and building projects together even before we had kids, so maybe we're unusually geeky? In any case, we still enjoy lots of playing as a family to this day, and for the record DS is a happy, well-adjusted, thoughtful 14yo kid who's incredibly self-motivated and independent-minded.
My mom rarely played with my siblings and me past the toddler years, and we turned out just fine. But this doesn't make the converse true, and I wouldn't want to see new parents become fearful of playing with their kids based on unfounded generalizations like this.