Um, no it doesn’t. The only thing the developer states is that it isn’t for backing up your home directories, even though it can do that if you add them. It just doesn’t do that by default, and that’s not its primary purpose either. Its primary purpose is to create a backup of your system.
So does timeshift. On my computer here, timeshift makes rsync copies to a partition on an internal spinning rust drive.
btrfs snapshots are not backups. They are rollback snapshots, so that you can roll back the content of the filesystem to an earlier stage. It’s a bit like version control.
In order to act on updates, you need a special pacman hook, but I have no experience with that.
It does however probably work on schedule, provided that you have cronie running. Most of the Manjaro distribution uses systemd timers, but timeshift requires cronie to be running instead (or alongside of it). You could of course convert the cronie jobs to systemd timers, but that’s going to take quite a bit more work, and then timeshift itself won’t have any control over that anymore.
Just because you don’t get to see the timeshift GUI doesn’t mean that it’s not executing a scheduled job in the background. The GUI is only for convenience, and is not started when timeshift executes a scheduled backup or snapshot.
It doesn’t have to. As the matter of fact, backups should never be stored on the same drive as the original files, because that would be pointless. If anything happens to the drive, you lose your backups.
btrfs snapshots are of course a different thing, although you can also store those on a separate device, if you know how. But then you’ll have to look at the manual, and I’m not sure whether timeshift itself supports storing btrfs snapshots on a separate device.
Either way, btrfs snapshots don’t take up much space, unless you keep a whole lot of them. btrfs is a copy-on-write filesystem, which means that only the modified extents will be copied, and the unmodified ones will be internally linked to.
Likewise, rsync backups also don’t take up as much space as you would think, because unmodified files are simply hard-linked against the initial backup, and only the modified files are copied over.