IMO the reason Kimbrel needs clean innings is because if he does not immediately get strike calls with that FB or even swing and miss with it to his satisfaction, he is going to keep throwing it regardless of the value of mixing pitches. He is going to keep throwing it until he is satisfied that he has it where he wants it to be.
If he has runners on that by itself can be a problem because he might walk somebody or at least the more memorable instances, give up a home run.
So he is fine coming into innings with runners on if that FB is terrific right out of the box or I should say to his satisfaction right out of the box. If not, he needs a few pitches and maybe even a batter or two to get in what he considers "his" groove. If he comes in with runners on, that is an issue. IMO, either he cures himself of that or you have to bring him into clean innings and give him room to work it out.