I go to games, and it is pretty obvious that the fans making the positive noise backing the team with chants etc are exactly the same fans who are chanting Rudkin out, sack the board etc. The rest of the ground essentially makes no noise whatsoever, aside from the occasional mild round of applause or tutting when there are mistakes. It's barely supportive or positive, more of a passive observer watching events unfold in silence.
There are no fans who "boo anything to do with city", they intermix positive team chants with anti board ones. Sustained booing of the team itself only tends to occur after games are already lost, mostly after some of the worst player performances in terms of effort the club has seen in recent history. And as proven in this thread, across Facebook and with incidents at the ground, there are many, many instances of "fans who desist from booing the club" verbally, or physically attacking fans who have booed or held anti board banners. You will find plenty of fans on Facebook demonising Union FS, or the movement to reform the Foxes Trust.