Plex is great, but not quite what I'm looking for. Ideally I'd like my entire collection to be synchronized and available locally on any device I use, and I'd like to be able to access the local collection through a web app (and have the web app be the thing that performs the sync'ing) without having to stream it from a remote source when the entire collection is sitting right there in my device's filesystem.

All of that is very achievable, except the "web app" part. I think it might be about time for me to just give up on that pipe dream and start building an Electron and/or React Native app instead.