OSR Apocalypse
I am tempted to take the 5E Wasteland books out for a walk. I have a couple but haven't gotten to use them. But after some thought, I think your original impulse was correct, that the stripped down, freewheeling, player-centric (rather than mechanics-centric) nature of an OSR game is the way to go.
After some poking about, I think that, if I were to run this, I'd use The Black Hack (2nd Edition). Do you know that one? It's very simple, but has some nuance, and I think it emulates the right parts of the Old School experience.
It also has the benefit of a ton of supplements. Classes, monsters, fighting styles, settings. All kinds of excellent material. There are even a few "Hacks" for post-apoc games (and one, Barbarians of the Ruined Earth, that explicitly cites Thundarr as an influence). We can use any or all of that material, or we can stick with the basic game to start and wing it from there.
If I were to run it, I'd be willing to commit to one "Mini Series" worth of content. Five or six adventures, each the length of one three to four session around a table (or about what would fit into a 44 minute TV show), mostly episodic with some kind of overarching plot, and probably a cool boss fight at the end. Might even do a 'Mid Season Hiatus" and jump a couple of levels just to see how the system operates at different points.
Assuming that all words and we get through it in one piece, we take a victory lap and then we either shake hands and walk away or we decide what's next.
Would that work for you?