There was a previous thread on the 218 (and related models) regarding hand problems that may shed some light on your broken wheel. I think the typical rust problem on these watches is your problem. You can see the previous thread old ref::here.
The minute pinion assembly (cannon pinion) is a two-part component on these watches. The actual pinion part is the part that drives the minute wheel. The larger wheel part of the assembly is driven by the gear train of the watch. When the hands are adjusted, the pinion turns within the clutch on its companion drive wheel. Is this the gear you said was stripped? The pinion part turns on the hollow idler post, and the two components (pinion and post) seize with rust. The minute pinion assembly must be removed, thoroughly pegged out, cleaned, the post glass brushed to remove the rust, and the idler post lubricated.
Since the minute wheel turns the pinion on the minute pinion assembly during hand setting, the minute wheel certainly could be damaged if the minute pinion had seized. That makes sense to me, but if you are shaking your head, I'll try again.
That is the minute wheel that is damaged. For that to happen, the pinion on part 579 (centre wheel assembly, somtimes called minute pinion assembly) must be frozen on the idler post. That minute wheel is quite robust, and I've never seen one go like that. Have you had the centre wheel assembly off? If it has seized, you will need to address that before you press a replacement part into service.