dfk41 I’ve restored a laptop from a Time Machine backup before. The good news is that it works great. The bad news is that it’s a somewhat slow process — slow enough that even if it were possible, trying to do that from a cloud-based Time Machine seems not to be feasible.
In addition, trying to backup to a cloud-based Time Machine would require the ability to upload a large amount of data at times. When I check in on what my Time Machine is doing during a backup, it’s not unusual to see over 1 GB of data going to Time Machine. My internet service has great speeds for downloading, but uploading is quite slow.
Personally, I have Time Machine set up to back up to a NAS, like you described. As far as the noise goes, it’s down in the basement, so I don’t hear it. Maybe another location is an option if you don’t want it behind the TV anymore.