Automated theorem provers are not new, in fact they are very old. One of the most automated is ACL2, which uses the well studied waterfall method (unrelated to waterfall development).

LLMs certainly use something similar, except they understand text as input. LLMs, especially used for marketing stunts, have way more computing power available than any theorem prover ever had. They probably do random restarts if a proof fails which amounts to partially brute forcing.

Lawrence Paulson correctly complained about some of the hype that Lean/LLMs are getting.

ACL2 even uses formulaic text output that describes the proof in human language, despite being all in Common Lisp and not a mythical clanker.

They do not think and use old and well established algorithms or perhaps novel ones that were added.