People don’t remember this now but the neocons weren’t not at all a fan of Kissinger and in fact viewed him to be too weak on communism . Many of them had supported Reagan in 76 precisely so Kissinger could be replaced by someone who was more Hawkish in general and in 1980 one for Ford’s demands to be VP was that Kissinger gets brought back and Reagan said no to that :
I don't think I ever want to meet someone who thought that Kissinger was insufficiently hawkish. What a terrifying thing to consider.
It's cause they opposed detente and thought he was too weak on communism. This group included people like Scoop Jackson and during the 1976 primaries the neocons went all in for Reagan as one core message of his 1976 campaign was the fact that he opposed detente.
Kissinger was not a neocon, and was pretty much the epitome of Realpolitck that so many neocons disliked.
You're not wrong. Kissinger's whole shtick was make friends with enemies, and preserve stability at all costs, even if it meant doing very harmful things.
The Neocons wanted a more DIRECTLY confrontational approach. Not the slimey behind the scenes realist approach.