Micheal and Trevor are old buddies from way back. There is obviously some deep friendship there, regardless of what happened after North Yankton.
While Franklin is new to the mix, there is friendship there too. Micheal obviously feels some connection with him, maybe father figure or whatever.
Anyway, should you choose the option to kill Trevor it doesn't go down well. Steve Haines orders Franklin to do this, and should he do so then Micheal will simply see this as being stabbed in the back by Franklin, no matter whether Trevor is a liability or not. Not surprising he would want nothing more to do with him after that.