Peter Weller is playing John Fredrick Paxton, a disciple of Colonel Green and former leader of Terra Prime. Go ahead, check Memory Alpha. John Harrison...who was an actual TOS crew member as someone else already pointed out...is somehow related to Paxton. I'm not sure how.
We've watched s4 of Enterprise.
It's just that nobody believes Weller will play the same character.
Exactly. Unless you also believe that, for example, St John Talbot, Chancellor Gorkon, and Gul Madred were in fact all the same person after some genetic surgery.