A default LineageOS installation will not be rooted. There are really less and less reasons why you'd even want a rooted phone. The killer feature used to be ad-blocking, but most ad-blockers can work without root nowadays.
The backup problem is a notorious problem with any Android phone, I don't understand what it has to do with custom ROMs.
"A default LineageOS installation" as in "a device which is officially supported through https://wiki.lineageos.org/devices/" is like that, indeed, and while most of the unofficial builds come with as -userdebug, no root, there's a high chance of encountering either a flaky build (permissive selinux) or having to fallback to a GSI build, some of them, and nowadays it means AndyCGYan's [0], come with optional root enabled.
As for backup: stock backup facilities (adb backup/restore) aren't exactly the best thing for sure, (rip Titanium Backup, Seedvault isn't even remotely there), and TWRP is broken on most of devices with modern encryption support (specifically, on Pixels). Coincidentally, Mediatek SoC based devices are the 'best' when talking about backup (as in: you can dump and clone all partitions through a built-in service mode [1]).
The backup problem is a notorious problem with any Android phone, I don't understand what it has to do with custom ROMs.