Best Disk Format for Time Machine on Monterey

ok, i misunderstood your question. but that new info brings up another issue in my mind. if the drive has been used for 5+ years, you may want to consider putting it aside incase you ever want some / all of the data on it and buy a new drive for your TM going forward. IMHO relying on a 5 year old HDD to back up your data is a disaster in the making. ALL drives will eventually fail. and considering how affordable they have become, there's no reason to not replace it.

if it's a new disk with no TM backups from a macOS prior to big sur, then it matters not wether it's formatted as HFS+ or APFS by you. any new drive assigned to TM duties will be automatically reformatted as APFS by the TM itself.

and, no, i don't recommend any apps that "clean up" "speed up" or "optimize" your mac.

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Other: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.

everyone will have a different opinion on that. :) but for me, yes "spinning rust" as you say is satisfactory for TM. i've never needed to use my backups (knock on wood) so speed is not an issue in my application. and of course, you'll save some cash over the price of an SSD.