I know the turn of phrase is popular even amongst biologists but I still think it’s weird to put it this way.
Evolution is a dynamic feedback loop on a multi-generational time scale. Plenty of neutral things can be transmitted for a long time without being culled out by evolutionary pressure and social behaviour can remain for a long time without being genetic at all.
It also shows who is best to lead a fight. Play much like curiosity, makes you able to navigate your environment. It very much has extreme usefulness.
To simplify a lot, either you have offsprings and your genes spread or you don’t and they don’t. Evolution is not a purposeful process towards fitness. It’s a reification of the results of the way genes are passed, how they and the environment affect individual characteristics and how an individual fitness to their current environment impacts their chance to mate.
Play hones reflexes. Its entire purpose is to train young animals on tactics, and on how to use their body, and on their local environment.
Put another way, if you aren't trained to use your body, you're more likely to die. The same goes with not learning tactics. Or what the local environment is like.
If you don't know what a tree is, if you don't know what a hole in the ground is, what a hill is, how well grass hides you or not, you are at a major disadvantage, if you're hunting, OR if you're hunted!
There can be other mechanisms to learn things, but play is one of them, and having children teach each other, lets the adult protect, and gather food to feed. It also ensures that youth is trained up on the current environment, not one that the parent recalls from youth.
Evolution is an abstraction subsuming other mechanisms. It doesn’t care about usefulness. That’s a finalist bias. Individuals reproduce or they don’t.
The rest of your post is pure conjecture. You can’t work backward saying: this thing is useful therefore it’s evolved. That doesn’t make sense especially for complex behaviour.
This has to be both right? Animals (and humans) evolved play because it has evolutionary benefits, but the immediate reward for play must be fun.