I totally get the frustration - emailing yourself files is definitely not it. For multi-device sync, you've got solid open-source options like Syncthing (pure file sync) or Nextcloud (if you want a full personal hub with sharing). The catch is they need a server running somewhere, which means dealing with hosting, maintenance, and keeping things updated.
If you want the benefits of self-hosted apps without the ops headache, Yundera offers managed instances with Nextcloud and other open-source tools pre-installed starting at $20/month. Full disclosure: I'm involved with Yundera. Check it out at https://yundera.com/ if you want the control of self-hosting without the burnout.
You've nailed the core problem: managed NextCloud with actual support is surprisingly hard to find. Disclosure: I'm involved with Yundera, and we built exactly what you're describing - NextCloud hosted and fully managed for you, with support included at $20/month, no admin work needed. It runs on open-source apps so you're not locked in, and we're based in France (EU servers, fast US connections). Check https://yundera.com/ if it sounds relevant, but the real point is your org shouldn't have to choose between ease-of-use and data sovereignty.