High Sierra has bricked 2 HDDs, now what?

High Sierras Disk Utility has one new 'feature', by default the sidebar in Disk Utility will only show Volumes and not Disks. To see the Disk click View in Disk Utilities menubar and then select Show All Devices, you should now see the Disk and the Volume(s) indented in the sidebar. To format the disk select the Disk not the Volume and select Erase and set the GUID Partition Map and Mac OS Extended (Journaled, same as you had done prior to High Sierra.

I think you did everything right, everything I would have done anyway. I bought a Seagate drive yesterday (not that exact model though) and had no trouble formatting it for use as a Time Machine backup.

Did what I always do i.e. erase with Disk Utility.

Do you recall the format you used? I used GUID Partition Map and Mac OS Extended (Journaled, Encrypted) as in Eau Rouge's instructions.

Whether the problem is due to Seagate's firmware or Disk Utility cannot be determined. In any event I'd try a different manufacturer until they sort out whatever's wrong.

I agree; I won't miss them one bit. Seagate isn't any better or worse than any other manufacturer. They're pretty much all cheap disposable junk. The only exception to that might have been Hitachi / HGST, whose drives were statistically more reliable than others, but they recently sold their drive division to Western Digital.

For what it's worth I generally purchase only bare drives and put them in enclosures myself, because most enclosures don't meet my requirements.

Edit to add: come to think of it the problems you describe might be due to the enclosure. I can't imagine the drive itself could cause those problems. That doesn't mean you shouldn't return it and try another manufacturer though.

Do you have another mac you can plug them in to?

Are you able to run the Terminal diskutil eraseDisk JHFS+ Untitled disk2 command? I'm using disk2 as your Seagate shown in a previous screenshot.