Just to add to this: offering managed SQLite services isn't really all that much of a thing because essential, as it is just a file, there isn't much to manage.

With databases like postgres you have daemons, config files and more that need an experienced DB-Admin to configure and run. All of that complexity grows exponentially when you include scaling on multiple machines.

With SQLite you essentially need backups. That's about it. A managed SQLite service would essentially just be something along the lines of:

"Tell us which file is your database, we backup it and if you need it restored press this button."