"Does CCC do a better job if moving everything to a new machine?"
I can't answer with assurance, because I've never used Time Machine to backup, not once, and will never use it. (I've even removed the Time Machine app and preference pane from my Mac).
I use ONLY "cloning apps".
But, yes -- I think a recent CCC cloned backup (you could also use SuperDuper) would be better, because I've read reports that when using setup assistant with a new machine, it will go faster.
I've seen numerous reports of trying to do a migration with a TM-based backup and setup assistant just.... hangs... towards the end -- it won't "complete".
I'm going to GUESS that a potential problem with trying to migrate from a TM backup (which may contain many copies of essentially "the same thing" over and over) results in setup assistant having to "pick through" each saved version of every file -- and then try to "sort it all out".
With a cloned backup there is none of this, as there is only one copy of every file to be migrated. Faster and easier to "sort out", because there's "nothing to sort"....