Time machine filling drive on first back up.

Since millions are using Time Machine successfully under Big Sur, it appears that what you have encountered is most likely due to something previously installed that was migrated over, or perhaps something newly installed.

A more successful migration over to a new system might result from a fresh erase/reinstall of a plain vanilla Big Sur and then just migrating over Mac accounts and files, but no settings, no software, nor anything else. Then test for recurrence of the problem, I expect it will not occur. Then you can reinstall your unique items a few at a time and retest to make sure the problem is not introduced. This may sounds like some work but I would do it in your shoes because what you are seeing with Time Machine is probably a symptom of a much bigger issue that will not go away but will only get worse.

I saw a "windows backup" in one of your partitions. Do you have a virtual machine running? If so, those virtual machines can generate large "updated" files on the fly and if that happened while Time Machine is running it will dutifully try to back up the "changed" file as it proceeds, and if this happens repeatedly it can use up massive storage on the Time Machine drive. In that case, simply excluding the virtual machine files or volumes from the Time Machine backup might solve the problem. If this is indeed what was happening, exclude all virtual machine files from the backups and instead, back them up separately and manually to a separate backup volume (not on the Time Machine drive). You can do that manual backup every week or few weeks.

If there is no explanation like the above virtual machine possibility, then I think the clean plain vanilla reinstall and then cautious migration only of files but of no software, settings, etc. is the path to follow.

What is the physical total space on the Seagate drive?

What about those other two backup volumes there -- how much space are they taking up? What is the space used and available on all those individual containers and volumes on the external drive? Are they all APFS?

By the way, Apple (in its documentation) recommends using a separate physical drive for the Time Machine backup, not sharing it with other volumes or containers, possibly to avoid interference from other storage on the physical drive.

How much free space do you have on your boot drive. If there's insufficient space to create a local snapshot on the boot drive TM will give a "not enough space" warning.