I can think of three options, none are perfect, but each could be tinkered with to get what you want:

In the Light of a Ghost Star – This is much further in the future than you intend. But it has incredibly simple rules which are easy to customize. Also I made a psionics supplement for it here. I had pretty good success running it.

Mutants & Machine-guns - A loose, fairly tongue-in-cheek system. Free.

Mutant Future - Basically a cleaned up clone of early edition Gamma World. Less light than other games mentioned.

I figured out the problem:

It wouldn’t let me post a comment-to-a-comment when the language was set to “Undetermined”. Setting the language of the individual post to “English” allowed it to work.

Weirdly this option wasn’t available to me on Jerboa. Only on the browser site.

I frickin love GURPS Ice Age supplement. But never have run it for two reasons:

1. After years of consideration I’ve come to the conclusion that I just don’t particularly like GURPS as a system.
2. I can never think of stone age scenarios that feel compelling in my mind. Not sure what quality would quite seem satisfying.

But still have some fascination with a dawn of humanity type setting.

Huh. Tried posting this response before, but Jerbo apparently ate my post. Turns out setting the language to English instead of Undetermined did thr trick.