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?
Looks super good! I like the contrast between the caning and the dark wood and the handles.
Which finish did you use?