Is there a way to have time machine work well with iCloud (keeping the backup there)?

iCloud is not a backup solution, it is a sync solution.
It will hold phone/pad backups, but not computer.

You cannot use it to store data that is not also present on your devices, other than such as offloading full-resolution photos or documents in low-space situations locally.

For true off-site backup you need to be looking at such as Backblaze.

The answer to your question is No.

There have been various attempts. I seem to remember a company which attempted to offer this several years ago as a commercial product (not even to iCloud, but Time Machine to "the cloud"), but I believe they discontinued it, and it was never particularly well-regarded in practice.

There have been lots of people who have suggested it, there have been lots of people who have said that it seems like something Apple should definitely do, but there is no way to do it at this point in time.

I believe that Arq can backup to your Time Capsule and can do hourly backups. It's not Time Machine, but it is, I believe, the next best thing.

I have found Arq much nicer to work with, and the developer actually responds to emails, which is more than you'd get with using Time Machine.

For history buffs, Apple offered a Backup app with a red umbrella icon at the turn of the century to back up select Mac files to iDisk for paid subscribers. The iDisk feature was first delivered on MacOS 9 (and people figured mods to get it to work partially on 8 and 7) and Apple clearly dropped this product over time after seeing how it worked and how supporting customers using it turned out.

• https://www.apple.com/newsroom/2000/01/05Apple-Unveils-Internet-Strategy/

iTools became .Mac which became MobileMe which became iCloud and here we are with Apple positioning Time Machine destinations away from local servers even. Over time, Mac backups have been transitioning away from Apple online services and towards direct attached storage.