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.
RPG
Discussion of table top roleplaying games.
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
I like TD2e for basically everything. They do have related systems/sets of Traits for different campaign types.
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.
Trying to figure out balefire in a D&D campaign sounds horrifying... would be fun though!
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.
Wow, I haven't heard of any of those. This decision is going to be harder than I thought! Thanks!