Glad you got your stuff back. One point of clarification. The migration you did, with Apple's help, was just that, a migration. Your stuff was moved from one drive to the other by Migration Assistant (I bet, since it was Apple), and that process can migrate your files and account information from just about any kind of drive. Given the CCC drive was a clone of your previous version, it migrated just fine. But you COULD have done a complete swap, including the OS, from that CCC drive if you wanted or needed to. Just boot from it and use CCC to clone it back to the internal drive (after formatting it to get rid of anything on it) and you would have had a completely new clone, including the OS. So the "sole purpose" as you called it, was still possible, but that isn't what you asked for help here to do. Basically, you used Migration Assistant to do what I recommended in Post #2.
Your second paragraph is a bit confusing to me. You said you made a clone of the 2014 iMac with CCC. Then you erased the TM backup and did another full backup. I presume you used Disk Utility to erase the TM backup as doing it through Finder is very, very slow and can often fail to delete some files because they are marked in the backup as system files. But using Disk Utility to erase the drive is faster and gets rid of everything on the drive. So, you erased the drive and did a new backup. Then you referred to "it" several times in this sentence.
Unfortunately, because the last backup from the CCC was 22 Dec last year it can't recognise the 'new' Time Machine and says that Time Machine isn't connected. In fact in the select disk pane are 2 identical Time Machines. I removed one and, when I started this reply it said that it would back up in 45 seconds. The attached shows what it says. I used to have 2 disks labelled Time Machine 1 & Time Machine 2. The second one died.
What is "it" that can't recognize the 'new' Time Machine backup? Is it the Time Machine app? Finder? CCC? What is the error message when "it" can't recognize the backup? What select pane are you talking about? Your attachment didn't attach, apparently, so we can't read what it says. CCC is not the tool to use to recover a TM backup, you use TM to do that (or Migration Assistant). CCC won't do anything with a TM backup. The migration you did should not have migrated Time Machine from the older backup to the new system because TM is considered a system application. So TM should be able to see the backup it just made. But I am wondering if it even made a backup at all, or if you erased the old one at all, if you are seeing two possible backups to choose from.
I used to have 2 disks labelled Time Machine 1 & Time Machine 2. The second one died.
When did that happen? To which one was the most recent backup sent?