Your question is actually three questions, with three separate answers.
1: Is it bonkers?
Yes. This can't happen the way you described. A solid won't come to rest midway down a liquid (or gas) that it otherwise sinks in (though Terry Pratchett has a beautiful section describing dead ships sailing underwater in the Diskworld series and there are liquids dense enough to float on), so your 'plates' won't float unsupported. If the whole crust is one shell surrounding the core of the planet then any kind of crack, fracture or instability would shatter it like an eggshell under a stiletto.
The second issue is the method of capture for these rocks: You describe the planet passing through an asteroid belt. Firstly it would have to be one hell of a dense asteroid belt to coat a gas giant in rock. Asteroid belts aren't the common trope of dense fields of rocky shards bouncing off each other. Even planetary ice rings don't achieve the kinds of density I think you'd need for this. Not only that: but these rocks are going to be hitting your gas giant at one heck of a speed: more than enough to punch down past whatever fragile equilibrium point you established.
Even if you manage to find a neutral buoyancy rock and have it suspended by thermal updrafts or stratified liquids or similar you hit another problem: Your human beings can't possibly survive. If the gas the plates are floating on is dense enough to support rock then either the gas above is dense enough to kill the humans, the humans are dense enough to sink through whatever makes up the plates or the plates are under a lot of pressure from below (which would tear the plate apart). In any of these cases humans die horribly.
2: What would need to be done differently to make it work?
Strangely enough this is a concept that I fleshed out for one of my first fully featured worlds, and it's a common trope throughout steampunk fiction or any kind of 'floating island' world where magic isn't allowed. Some form of element that is a room temperature superconductor. In your case specifically it would be held up by magnetic locking in the magnetosphere of the planet's metallicised hydrogen core. If this element is common and ubiquitous enough in the crust of the planet then you can fine tune the height of the plates in the atmosphere. I'm fairly sure there is another question on WB about whether there is an idea height for this, so have a look around and see what you can find.
3: Are there better ways of achieving my initial desire?
Yes, if your 'large' world is only a little large (so not a gas giant) and you're willing to give up on a totally thin crust in favour of a very 'pocketed' one.
If instead of a thin crust hanging over nothing you go with a cavern riddled planet with rare but violent bouts of vulcanism you can achieve poisonous gases, creatures living in (fairly small but still hostile) subterranean gas environments and violent rearrangements of the world that can literally sink whole continents and cause new ones to be pushed up from under the ocean, forming new pumice-stone-like continents for both the human populations and the subterranean to migrate to. This isn't exactly what you asked for, but hopefully it sparks a couple of ideas (in fact it's got me thinking of a question I'd like to ask...)
Happy world building!