I really liked the first two episodes (alternative timeline episodes) with Kirk. But more Kirk, that isn’t sporadic or kinda in the background, is essentially a distraction from the SNW crew IMO.
Something I can’t unsee with Kirk being prominent in SNW is the Riker/Troi appearance in the last episode of Enterprise … this sort of fan-servicy cameo that kinda makes sense but in reality is an invasion of a series by the bigger starts of another series. That scene in the last Kirk episode where it paused on Uhura, Spock and Kirk sitting around a table together for the first time, while wonderful, really felt like an insult to SNW, not to mention that it didn’t actually contribute anything to their relationships/characters in TOS.