Sync (iCloud) vs backup (Time Machine) - how to do both?

Being relatively new to the Apple world, I want to be sure that I have this straight:

I have hundreds of GB of data, so storing everything on my Mac Mini m4 internal drive is not an option. All that data is instead on a fast external drive. From there, I can back up everything using Time Machine to a spare drive. So far, so good.

I also want to sync the data to iCloud Drive (or any other sync service for that matter). But iCloud and all of the other sync services I've seen use the Mac File Provider interface, and that, so far as I can tell, caches files in a library on the internal drive – which rapidly fills up. Not so good.

If I understand correctly, though, iCloud can adjust storage so that the most frequently used files are stored locally while the rest remain in the cloud. That, in turn, means that access to the folders/files that don't happen to be stored locally would be pretty slow, although maybe I could live with that. What I'm really worried about is not knowing if I can back up all those hundreds of GB in the cloud using Time Machine – that does not appear to be possible.

Clarifications: please don't suggest that syncing is the same as backing up. I've already lost data (nothing disastrous, thank goodness) that was synced but disappeared for some reason; I also want versioning, which is provided by backups but not syncing. Also, backup doesn't have to be via Time Machine, in fact a could-based back would be fine -- but iCloud is not that.

So am I understanding the situation correctly? Surely others have addressed this issue? Thanks in advance for any insight!