the timer has no idea if it was triggered during last boot. It only has the context of “this” boot, so it will do it right after a reboot and set a timer to start the service again after a week of uptime.

This is not correct. Persistent=true saves the last time the timer was run on disk. From the systemd.timer man page:

Takes a boolean argument. If true, the time when the service unit was last triggered is stored on disk. When the timer is activated, the service unit is triggered immediately if it would have been triggered at least once during the time when the timer was inactive.

OP needs to remove Requires=backup.service from the [Unit] section so it stops running it when it start the timer on boot.

You have the timer requiring backup.service, so it will run that service every time the timer starts on boot. Remove Requires=backup.service, and that will fix the issue.

Well, for one, it’s network attached storage. If it’s not present in the network for one reason or another, guess what, your OS doesn’t boot… or it errors during boot, depending on how the kernel was compiled and what switches your bootloader sends to the kernel during boot.

Just use nofail in the fstab.

Second, this is an easy way for malware to spread, especially if it’s set to run after user logon.

If your fileshare is accessible to you, it is also accessible to malware running as your user. Mounting the share via a filemanager doesn’t change this.