Given that this "reels in" useful functionality in preparation for a new design, I think I'd want an actual plan for what that design is prior to merging this. Otherwise how do we know whether or not these changes are enabling the right things?
I'm also not convinced that these changes make future changes much easier. Anything that changes how stages work will need to touch the majority of these lines again.

The next step moving forward would most likely be to relocate run criteria storage and eventually evaluation into Schedule, resolving issues around not being able to use a state from multiple stages, and having hard sync points within blocks controlled by run criteria. @Ratysz might have further insight here.

I'm pretty confident having a locked-in stage topology will make moving run criteria to a central location easier, which is why I'm doing this PR first. We could postpone until more work is done, but that might lead to a rather bloated PR that's tricky to review.

I think existence of trait Stage is largely incosequential for all changes up to the one that deprecates stages as a concept. We'll have to contend with them one way or another up until we remove them entirely, trait or not.

Closed in favor of waiting for bigger changes with #2801