whoops, meant to write “books” not "movies!
I use MediaTracker, with the plugin for Jellyfin to track progress. It works pretty well. Kinda barebones and I’d like to see some creature comforts added, but the dev is busy (and they try to be responsive as best they can). I like that it is fully open source and lightweight. I almost did flox, but wanted movies and games too.
I prefer not to sync my information with Google or other entities either, but do not think that using Linux precludes using whatever tools they like. I don’t think this is a question about ideals, but about tools and compatibility.
I personally also doubt that Google Tasks supports that capability, unfortunately. I personally use Thuderbird for email on Linux, and it supports tasks too (I use Nextcloud for my tasks but I bet it supports Google as well). Not the same as your gnome extension, but an option.
Good luck!