Mutants & Masterminds 3e – available for free online as d20herosrd.com – can do literally anything. It’s originally written for superheroes (obviously), but I’ve seen it used for D&D style fantasy, sci-fi, and so on. There is literally one single spell in all of fiction and film that I have tried to replicate in that system and not been able to (Balefire, from The Wheel of Time, because weird shit with time). It’s all the flexibility of GURPS with a fraction as much math, all the power-fantasy of D&D without the game-breaking power-loops (there are a few but they’re easy to spot and avoid). And all the complexity is front-loaded; once you get through character creation, it’s all just d20s. If you can play D&D you can play M&M.

For something less crunchy, try Fate. Whether Core or Accelerated or whatever I couldn’t tell you – I personally hate this entire family of systems – but it is very rules-light and storytelling-focused. It might suit your needs.

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 like TD2e for basically everything. They do have related systems/sets of Traits for different campaign types.

Recently I’ve tried Blades in the Dark which from my perspective ticks a lot of your boxes. Simpler than D&D for sure, similar to PbtA in terms of rolls and character sheets, and set in a post-apocalyptic world (albeit Victorian era).

Other than that I love the flexibility of Savage Worlds with its kind of simple rules and many many different extensions for different settings. Good base for tinkering imho.

And of course if you mention PbtA, why not go with Apocalypse World?

@Hollow_knight I did my last P-A game with GURPS, which unless radically stripped down ain’t exactly kid-friendly. What’s your core story in this game? Crafting and skill trees sounds almost as complex as 5E, especially given that the kids already know the latter… Might have a look at some of the simpler derivates of 5E (Five Torches Deep, Shadowdark), create a few custom classes…

I’ve never been a big fan of PbtA, so I’m not a big help there. Within that rules genre, I’d have a look at FATE Condensed plus the cliffhanger mechanic from the Masters of Umdaar setting (PDF available for free).

Story over numbercrunching makes Fate a great choice, particularly Accelerated, which is a streamlined minimalist version. Minimal character stats, simple rules, and it’s very flexible, with a lot of the specifics being determined during play to fit the plot. For example, instead of specific skills you have Approaches, which describe how you do things. Your character might be good at being Clever, or Sneaky, or Forceful, etc. You get a bonus to the thing you’re doing if you can describe how you’re doing it to fit that adjective. It’s a system designed to be more of a collaborative storytelling game than a wargaming system, which sounds like it’d suit you.

Flatpak

Flatpack: Fix the Future is the first game in the smash Hyperoptimistic Postapocalyptic genre of gaming. Flatpack is a game about building a new society using pre-apocalypse technology that you don’t always trust or understand, and solving terrible problems creatively. 

This is not a complete suggestion, but based on the setting, you may be able to take some elements from Gamma World.

How simple do you want to get? You could reduce PbtA down to a single move Defy Danger. Still works.

Apocalypse World is a pretty good one. It’s very rules-lite and narrative focused. Players are more in control of the narrative than most RPGs; the players can declare what they want to do (e.g. “I want to drive really fast towards that group of bandits, then jump out of the truck just before it hits them, do a backflip in the air, and shoot the two on the left while the truck runs over the two on the right”), and rather than the response being “You can’t do that”, it’s typically “Okay, but”. Getting a die roll that isn’t high enough to just flat out accomplish what you were trying typically still lets you succeed, but with some caveats, which the player typically gets to make the final decision on - for instance, in the above case, you might roll a 7 (a middle-tier roll), and be told that you can choose 2 of the following: You accurately shoot the two bandits on the left, the truck kills the two on the right, you don’t hurt yourself when you land, and your gun doesn’t explode.

I think Barbarians of the Aftermath could be a good fit for you. It allows to make up your own apocalypse, to customize it as you like. And it based on the simple and excellent Barbarians of Lemuria system.

Sorcery and Super Science might work for this. It’s already most of the way there. Most characters are mutants though and all humans are sorcerers so it might not be a perfect fit

You can also try a bit of a combo move with Worlds Without Number and Stars Without Number. Both games could probably handle this by themselves, but the rules are similar enough to combine which would give you scifi and magic together

Apocalypse World 2.0/Burned Over is pretty good. Easy systems and the creator revised it to be more appropriate for his own kids, if I’m recalling correctly.