Based upon detail from another forum ( see link ), which although I'm unfamiliar, a post sheds light on Monterey & Spotlight indexing pertaining to Time Machine external drives. {I'm using a 2020 Intel iMac & now, macOS 12.2 & a 2TB SanDisk Extreme Pro SSD, that has been in continuous use since 2019}
When I attempted adding my external Time Machine drive, I received an error message... "xxx is a Time Machine backup folder. You can not add it to the privacy list." This corroborates what the forum's link points out... Monterey's Spotlight does not index network drives. (Although the guy in that forum wanted to index his Music Library on an external drive & figured out a Terminal procedure to do so.)
I've been using my SanDisk SSD for Time Machine without issue, until Monterey. As I previously did have my SanDisk SSD added to the Privacy window in pre-Monterey Spotlight, I deleted the SanDisk SSD as part of the process getting Time Machine functioning as it should (& as it did prior to Monterey).
Let me add a couple of recommends which might help others... a caveat to those using newer, M1 Macs.
1. My SanDisk SSD (and, Time Machine) was rendered unusable post, Monterey install. I found the best way to erase the SSD (essentially starting over) was by erasing the top-level disk, using Monterey's Disk Utility, and reset the SSD to Untitled, ExFAT, & Master Boot Record. [Note: I was very fortunate I had a good, Carbon Copy Clone on another SanDisk SSD - so, my loss of all prior, Time Machine backups was not a big problem as I could not get access to any of that backup Data.] ...Then, I erased the nested drive to Untitled, Mac OS Extended (Journaled), GUID Partition Map; and then, Converted the nested drive to... APFS. ...Once conversion was complete, I used Get Info to rename it as I wanted. ...I can say that with the completion of this process, I no longer experienced any issues mounting & dismounting (using Disk Utility) the SSD. When I re-selected the newly converted, SSD for Time Machine, it set up & completed the 1st backup quickly. Time Machine backups have since been running smoothly.
2. I have found macOS Disk Utility has been the more reliable way to dismount an external SSD, instead of using Finder. ...And as result, I use a created, Automator app that employs a Terminal command {to Disk Utility} for dismounting my Time Machine SSD. {The SSD gets too hot to leave continuously connected; a pain, but so it goes.} ...I keep this Automator app in my Dock for easy access to dismount the Time Machine SSD.
Having employed the above for the last 5 days (from this Post), I'm confident the external SSD, Time Machine drive will continue running well. ...When the next macOS update takes place, I will find out if the current, Time Machine set-up transitions & works with the newly updated, Monterey, or the Update causes problems {meaning, hasn't yet resolved the basic, Time Machine issue with external SSDs}. ...And, we'll have to again sort out Time Machine.