Time Machine was certainly broken after Sonoma. I have relied on TM for such a long time, and bought a brand new M2 Mac, but it came with Ventura. Apple support recommended I upgrade to Sonoma, and I did, and in March 2024, had my first ever issue with Time Machine. Time machine would be laggard and hang and then eventually fail. I did not realized it identified the file it could not process, but then realized it was just a .png file I had copied into notes.
I went to 'finder' and deleted the .png file removing it from trash as well and then backed up successfully albeit slow.
I did a LOT of research / google etc and found was not alone. Having 2 Mac Air's M1 and M2, I did not update the M1 which was still running Ventura. I went and bought a new Hard Drive (LaCie Rugged) and formated that with APFS as the other was on Extended journal and ran 5 backups on that new drive. I then reformatted the Extended journal drive erasing and reformatting and adding to TM and then ran backups and they have been flawless.
I do belive that something in Sonoma IOS 14 croaked the Extended Journal file system making it slow and very troublesome. I now have 2 drives for each of my Mac Air's (M1 and M2) and all drives are formated with APFS (Apple file system).
If you are unsure how to delete and reformat to APFS (Apple File System) give apple support a call and they will assist.
Sonoma I think, has a lot of overhead, but a simple .PNG file should not prevent Time Machine from completing and i've used .PNG files for years in my "notes' when planning trips and copying screen shots of places, and all of a sudden Sonoma took away my confidence in Time Machine, which I've used as my 'life line'
But as I wrote this, I did upgrade my M1 to Sonoma 14.4.1 today 5/8/2024 and when running the 1st Time Machine backup on the reformatted drive which has several Ventura Backups, the 1st Sonoma 14.4.1 it seems all is well now with Time Machine.
I am however, disappointed that most of the Technicians at Apple were totally unaware of the Sonoma and Time Machine issues and there certainly are a lot of stories out there about people having issues.
Billg76