Is there a backup tool that the Elementary team recommends, even though they do not directly support the software? Or is there a technique within the OS I have yet to discover? There is talk of Timeshare in the responses I have seen but I cannot find it in the App Centre.
My tool of choice is Back in Time. It's very versatile (incremental backups, local/cloud, enrypted/non encrypted etc.) and has a clear GUI that allows you to keep track of your backups and restore (complete folders or single files, as you like) very easily. It should be in the repositories if I remember correctly. If not, check here: https://github.com/bit-team/backintime
I'm not sure what the elementary developers would recommend, but I can make a personal recommendation for a nice GTK3 back up app called Backups. It's in the AppCenter and can be easily installed and configured. I use it to back up my /home directory to an external drive, that then gets pushed once a week to Google Drive via a cron job.
There are other backup options out there, but personally I've found Backups to work well for my needs.
Best of luck to you!