I would say the opposite. Ente has one huge advantage and that it is e2ee so it's a must if you are hosting someone else photos. But if you are planning to run something on your server/NAS for yourself then Immich has many advantages (that often relate to the e2ee). For example... your files are still files on the disk so less worry about something unrecoverably breaking. And you can add external locations. With Ente it is just about backing up your phone photos. Immich works pretty well as camera photo organizer.

Does it have a mobile app that backs up the photos while in the background and can essentially be "forgotten"? That's pretty much what I need for my family: their photos need to get to my server magically.

I'm also a very happy Ente user. I use Garage for its S3-like storage, with one of the nodes running on my local network (LAN). My local DNS (CoreDNS) is also configured to use this local node for the domain, which makes everything very fast.

I'm a very happy Ente Photos user as well.