Which Linux Mint Timeshift method is more space efficient

Can I expect a difference in drive space used for backups between the btrfs option and the rsync option?

Yes. The rsync option uses hard-links. This means that if a file remains the same between snapshots (these are not backups, BTW) the file system only needs to store a new name/path for the same file. This also means that if a file changes between snapshots the entire new file needs to be copied.

In contrast, BTRFS snapshots work at the file block level. This means that if a file changes between snapshots only the changed blocks need to be retained in the new snapshot; The other blocks of the same file can be shared between multiple snapshots.

Another thing which affects the amount of space used the filesystem overhead, which varies between filesystems.

Are there any drawbacks to using btrfs instead of ext4 for a home system?

Yes. Before using BTRFS it's worth reading the documentation. This is not your everyday filesystem. In particular read the Gotchas which discusses fragmentation of random-write files (such as Firefox sqlite databases).

If some kernel update or whatever did prevent the system from booting, will Timeshift pop up during boot as a recovery option?

In short, I don't know. Timeshift does integrate with the bootloader, but I don't know to what extent. You may need to get with the devs on this and do some testing on your own.

It's too late but maybe someone can read.

For the 3rd unanswered part.

As you asked in the comment, it doesn't come with the system so it won't pop up.

To back up, make a bootable and install a timeshift in bootable (during Linux install, in the bootable mode you can access files.) Install timeshift by whatever method you like (apt or whatever). Open timeshift and then restore.