I discovered that if I select "Sleep" from the Apple menu for our new M1 2021 iMac, our hourly Time Machine backups work fine even while the iMac is asleep. So I set "Turn display off after" to "Never", set my Screen Saver to the "Message" screen saver with spaces for a message, and set "Show screen saver after" to 10 minutes. So I now have a low impact screen saver if I'm going to be away for a little while and the "Sleep" function if I'm going to be away for a longer spell.
This is different from our older iMacs, where we can separately specify when the display turns off versus when the hard disks turn off (so we can automatically turn off the display after 10 minutes and the hard disks after 3 hours) and never have Time Machine backup problems even on older Iomege Firewire disk drives. On the older iMacs, we routinely turn the display off with Control+Shift+Eject when we are stepping away for a little while.
I can't find any clear documentation on exactly what is supposed to happen with Monterey 12.3 on an M1 2021 iMac when the "Sleep" function is invoked, versus what happens when the display is turned off automatically after a specified time period. (After talking with Apple Support, I still don't have a clear idea.) But given my experience, there is something odd going on with either the M1 2021 iMac on Monterey 12.3 or our older Firewire 800 Iomega disk drives that causes Time Machine hourly backups to fail on backing up certain selected files if the display has gone to sleep after a specified time period.