Why not just make your own depot, install all the packages you need within there (and instantiate or not depending on whether you want it to be tiny or huge), tar+gz or zip it and send it off to students? I don’t think you’re going to get around doing the initial precompile of all packages, but how much time does that really buy you anyway? 5 minutes maybe?
Edit: Note that making your own depot is as simple as running JULIA_DEPOT_PATH=/my/depot/path julia and running ] up.