From what you’ve written, it is unclear to me which aspects of DnD you want to preserve and which modifications PbtA would bring to the table that you are looking for.
Why do you avoid DnD in the first place? What is the basis for your discomfort? From the mechanics, I would like to say that 5e is very well rounded and has to offer a good experience.
On the scale of crunchy systems it is in the lower third, having Cthulhu dark on the lower end and Shadowrun or Pathfinder on the higher end.
What you get baked into the system is combat in terms of mechanics and high fantasy.
What I really like about 5e are things like the advantage rolls.
In comparison to dungeon world combat looks like a minigame within DnD but that is due to its heritage.
These are mechanical things which are IMHO essential to offer the DnD experience.
That is comparable with the core move mechanic including the typical 2d6 for PbtA. You destroy the PbtA-Experience when you start relying on skill checks.
If you want to preserve DnD you have to go with the mechanics. From my perspective Dungeon World is only part of the spirit of DnD, but represents more of what I am looking for in that type of games.
That said, what would be possible within the DnD spectrum is running the game more story focused. The prompt “What do you do” works of course as well as the concept of fronts and dangers. There are no rules for how the story changes after a skill check was made, but nobody holds you back from interpreting the result and make a soft or even a hard move - only that there is of course no concept for that, but you can look at it as a codex for the GM. You could run dungeon starters, ask questions and play more player facing.
You are free to houserule resources including gold and things your adventurers carry with them. If it makes sense they could have a rope when they need one - perhaps you play out the flashback how they prepared for such a situation. Nobody holds you back.
Omit XP and grant level ups regularly and you stop your party from pea counting and calculating through every encounter. Although this might be a hard modification for some.
Tweak initiative: play with group initiative or Greyhawk initiative etc. But in comparison to other systems combat is relatively streamlined (of course not comparable to PbtA).
Last but not least forget about alignment and CR / balancing.
I think it is easier for you to streamline the playstyle of DnD than to find a corresponding PbtA spinoff.
Dungeon World brought a lot of inspiration for DnD GMs.