Time Machine Available Space Size Discrepancy

Unless you are using the whole NAS capacity as a TM backup, you probably have TM set to back up to a volume on the NAS...what is the size of that volume? Alternatively, most NAS software allows a user quota to be set and that could also limit the space available.

Even so, the above may not be the whole story since in any case TM should delete old backups to make space available.

Time machine does not create its backup files on the bare metal of the NAS drive, which may be in a foreign and possibly even unwritable format.

Time machine backups are stored in a sparse bundle disk image file. That is essentially an entire Mac file system inside one file on the NAS.

You also mentioned that you had started running Sequoia, which complicates matters enormously. When you do a major upgrade, your Mac (rarely) appears to be a different computer after the upgrade, requiring a different backup destination.

Even if that is not the case, that amount of stuff you need to save, including all new System files, is be quite large, and it simply may not fit inside the sparse bundle disk image file.

In some cases, you can reduce the required size of the backup dramatically if you deliberately add the folder named "System" to Time Machine exclusion list. When you add it, you will get a dialog asking if you intend to exclude all System files, which you should do. Because System files can not be used to create a usable locked system volume, you may as well exclude them. (Now we re-install MacOS itself from fundamental sources, NOT backups.)

I created a new TM back up to the NAS and it shows 12.29GB. The backup runs for hours and says that it has copied much more than 12.29GB, but eventually fails saying not enough space.

I would suggest creating a dedicated space of a fixed size for your TM backup destination on the NAS. Apple recommends ≥2x the size of the drive being backed up. For example, I have a similar setup on my NAS with one volume and multiple shares, I have five Macs backing up to it and each of them has a dedicated shared folder with a quote of 2x the internal SSD size, with the remainder of the volume for general storage.