You’re exactly right that the run folder has the raw files and configuration to actually serve the site.
I’m not 100% sure what would happen if you moved that and symlinked it. It might work since you’re on a more unix-y system – Windows users would likely be in a world of hurt.
I’d say give it a shot by:
Take a backup of each site by right-clicking on it in Local and selecting “Export.” That will give you a zip to save in case something goes wrong.
Stop Local
Make the changes to that run folder – symlinking etc.