When Columbo discovers Lenore Kennicut's relationship with Archer, he mentions this to Brimmer as a possible motive. Then he spends time trying, and failing, to chase down Leo Gentry.
Why doesn't he just ask Arthur Kennicut? Then the falsified report would quickly become obvious and Brimmer would immediately be under suspicion.
Further, why doesn't Kennicut volunteer the report right at the start? He has no reason to withhold it. He was given the report ON THE SAME DAY as the murder - surely he must have thought it relevant? Instead Columbo just accepts "clean bill of health" and "a personal matter" then digs into his remark about "sports" for no obvious reason.
Arthur suspecting his wife of cheating could be considered a motive. Maybe he wanted to keep the hiring of a PI and the report out of the equation for the Lt. at the beginning, since Arthur really WASN'T the killer. This would just raise unnecessary alarms and AK wanted the killer found ASAP. Plus if the report was perceived by Arthur as truly a "clean bill of health", it may have been considered totally irrelevant in his eyes to introduce this report into the case.
One thing is certain. Brimmer went way out of his way to get Columbo off the case by offering hom the job. I think - and others on here do too that Brimmer is guilty of manslaughter. Not murder.
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."
In many cases throughout the series we see suspects offering "too much" help to try and take suspicion off them. For instance:
Justin Rowe and Cooper (Columbo Goes to College)
The Commissioner (a Friend in Deed)
Abagail Mitchell (Try and Catch Me)
The list goes on while Columbo continues to give them rope to hang themselves.
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!