They did have that mini-arc with his drug addicted daughter (that hated him) who the police forced to spy for them, and she got killed. We could have skipped that and not lost anything.
OTOH, the more I learn about NYC I really want to know where the characters live. And how do they afford it if they live in Manhattan? Apparently Logan still lived in Manhattan all the time he was working in Staten Island.
. . . but, what it did allow is 79 hour-long (48 mins or whatever it comes out to) episodes worth of character and relationship development to unfold in a very slow manner. I think it’s interesting that, while the plot of TOS is totally serial, the relationships grow to be a sum of all the individual parts.
That show might have been interesting with a true arc, but I am pretty fond of the impressionistic/pointillistic way the characters grew out of series.
Also, I think it laid a great foundation for ST II-IV.
I’m very thankful that although I’m pretty sure Columbo was married, I don’t know anything about his wife, his kids, or his home life. And oh, by the way, I don’t care about his relationship with the boys down at the precinct, either.
So, you never watched Mrs. Columbo, I’m assuming? 
I have wondered about his relationship with the other officers. He never has a partner, nor any true subordinates. Doesn’t an LAPD Lieutenant usually command a squad? But he’s always out there detecting on his own.
Even more, I wonder what the other detectives think of him. Do they think he’s a nut (a nut with a high closing percentage, though!)? Or does he act “normal” around the other cops, and only acts scatterbrained while talking to the suspects?
Ironically one of the few times Star Trek TOS referenced past episodes overtly was in the last one, Turnabout Intruder.
I agree we could have skipped it and not lost anything from the episodes it was part of, or the show in total, but I do want to say that Jerry Orbach turned out some really good acting in those scenes, and even in his one little portion, so did Benjamin Bratt. When Lennie starts crying (what look like real tears) and says “She was my baby, what am I going to do now?” and Rey puts his arm around Briscoe and replies “You’re coming home with me, partner.” The fact that they have never been especially close makes it even more touching, and I tear up every time. Actually, the story arc wasn’t that big a part of the eps in was in, and they make sense even when shown out of order. It just goes to show that less is more.
I don’t know about Logan, but I suspect that Briscoe was in some rent controlled apartment. He’s been divorced for a long time, and he may have rented a two bedroom 20 years earlier, which would be the 1970s. He could have gotten an apartment in some place like Columbia Heights in Brooklyn which was right by the Brooklyn Bridge, and would have been a short commute to Midtown, which is where it seems the 27th precinct is. Columbia Heights was very cheap in the 1970s, and a lot of people who worked in Manhattan lived there. My aunt and uncle lived there for a while, and still got their kids to a private day school in Manhattan, the same one I went to and we lived in Morningside Heights, the area between Harlem and the Upper West Side. We lived in the cheap part.
Staten Island is not a good place to live from what I understand (I have never lived or worked there, and I have either lived or worked, or both, in all the other boroughs). If Logan has a rent controlled place, and it is close to a subway line, he is better off keeping it, especially if he harbors some hope of going back to Manhattan, or at least going some place like Brooklyn or Queens. His family has been in New York for a few generations, so it’s possible he inherited a place, because it’s true he’s younger than Briscoe, and unlikely he got one on his own-- I mean, a really good one worth hanging onto.
It’s also possible that he had a rent controlled place he sublet. It is legal to do that as long as you have the landlord’s permission.
Oh my god! They killed Kenny!
Chances are they don’t live in Manhattan and there’s a pretty good chance they don’t live in the five boroughs at all. This is fairly recent, but things have probably not changed that much in the past 20-30 years. There have been a lot of NYPD officers living outside the city since the residency requirements were relaxed sometime in the sixties.
George Carlin: We called our little neighborhood “White Harlem”. Because it sounded bad, you know. It’s real name was Morningside Heights.
RivkahChaya, thanks for the info!
One thing:
because of crime, or because it smells like a landfill?
Because otherwise, how many times can you kill Kenny? Just once. 
Back to OP, there’s “Continuity”, and there’s “continuity”. Big continuity is arc-based writing. But little continuity is simple consistency in characterization and the recollection that some things happened in the past and can have an effect on the future (i.e., being mentioned), even if they’re not critical story arcs.
Big-C Continuity: Babylon 5. Battlestar Galactica reboot. To a smaller extent, The Simpsons (Sideshow Bob sort-of-story-arc).
Little-C continuity: almost every episodic show has this to some extent, even if only to the degree of not arbitrarily switching around characters or erasing major life changes. Lucy Ricardo was pregnant, gave birth, and had Little Ricky. In that order. And consistently. Even if the episodes of I Love Lucy themselves were mostly self-contained.
This reminded me of “No-Continuity”. In McMillan and Wife, Sally was pregnant twice, yet no children were ever seen. Not once, not ever. Nor any evidence that a kid lived in the house. And when Sally died in the plane crash, only one son died with her.
I’m not sure that’s a better or worse way to hide an actresses real-life pregnancy as opposed to having her stand behind things and hold pillows and whatnot.
And every episode one of the crew would ask, “how can that be, Captain?”, as if he knew any more about it than they did.
Yeah, but Ace was a) only with the 7th Doctor b) not killed at all.