When this thread got started, I also noticed in the popular press that Synology NAS devices were vulnerable to remote attackers. See the latest update: https://www.synology.com/en-global/security/advisory/Synology_SA_24_27 . I don't use Synology NAS, but just making the observation.
In my case, I'm doing backups in my home wi-fi LAN, and I'm doing backups from one Apple device (MacBook Pro (Mac15,7) [Source is Mac SSD internal] to 2 different destinations: 1) WD hard drive attached to an old Airport Extreme via USB-A, and 2) Mac Mini (Mac16,10) with a Samsung T7 SSD connected via USB-C. My source drive contains about 770 GB of files to be backed up via wi-fi network.
Like many others in this thread, my Time Machine (TM) backups started failing after upgrading to MacOS 15.0. The reasons for failure were sometimes due to a network disconnect by my MBP or to the remote destination device not cooperating with TM.
I've had problems of TM backups failing after OS upgrades a number of times in the past 12 years with Apple products. Generally speaking, I've had the best success by creating a new backup (after first moving the old backup destination files to a different device, so I could restore from it if necessary). The new backup can take days to complete, so I do what is needed to speed that up (since failures would often happen after many hours had expired).
After the first full backup was successfully completed, then I changed my Lock Screen settings back to what I normally prefer (on both source and destination).
Most recently I changed Lock Screen settings while keeping the MBP connected to power with lid open, so it wouldn't go to sleep during the TM backup. Likewise for the destination machine (Mini) and its external drive. I manually started the backup, which took about 22 hours for about 750 GB, not letting either the source or destination sleep.
Also, it seems that, over the years, MacOS has changed some of the instructions for TM, depending on the OS version, especially in how to prepare the destination. I have found it worthwhile to read the Apple Support docs for the current OS version, and checking all of the various settings (e.g., especially for file sharing on the destination and settings for folders for TM backups).
Sometimes my antivirus software has interfered (BitDefender) and I have set my Time Machine protection to Off.
I have not changed MacOS firewall settings on either source or destination. I keep the firewalls on.
Yeah, this is annoying and takes time to debug. There's lots of places where problems can occur, and that's what makes it so perplexing and frustrating at times. I also use other backup strategies besides TM for really important files, using external cloud services for those.