Yeah, Brimmer's arrogance is too high. First he shows up on Kennicut's behalf, since the Lt. mentions it was going to be slow. Then he straight-up offers Columbo a job, and when the Lt. fake bites and asks if he'd still be on the Lenore case, Brimmer says no. This was a dead giveaway. Not too bright.
Makes you think back to when the Lt. overtly says to a frustrated Arthur that he's sorta stuck and not sure where to go next (shakes the tree, knowing AK is a big wig news guy with associates, etc), and after that scene Brimmer appears with Arthur. Just what the Lt. wanted. hehe
"You know, I suddenly feel very much more optimistic about this whole thing.
Uh, it's not based on anything, no facts, but, you know, I'm a superstitious guy. You know, I believe in signs. I believe in palmistry and astrology and all that kinda thing.. YOU don't, I know that."
"Oh, definitely not."
"Let me see your hand."
I don't think Columbo was expecting Brimmer, or anyone, to suddenly show up at Kennicut's home. But, Brimmer showing up and offering his help out of the blue, gave Columbo pause to wonder. Kennicut didn't go to Brimmer for help, Brimmer went to Kennicut and offered his help. They weren't close friends and all we know, their association is just the one "personal matter". In Columbo's experience, people volunteering help with a case, usually means they know than they are letting on. And since he doesn't have a suspect or any leads, as he tells Kennicut, he starts right there with those two men, to either rule them out or rule them in, and so he checks their rings.
In the movie COMPULSION, the fictional version of Richard Loeb (played by Bradford Dillman) keeps trying to help the police since he knew the victim and knows the neighborhood. I don't know many true crime stories remotely well, but I've heard that Richard Loeb really did do that!