Has there ever been a TV series where the title sequence was just part of the episode (see explanation)

Has there ever been a TV series where the title sequence flows as a completely natural continuation of the cold open.

What I mean is, for example, in Popular Weekly Space Opera, the cold open takes place in spacedock in orbit of Earth and the captain of the starship Hubris is bemoaning the delays in the repair schedule for the ship, when he is interrupted by an important call from the fleet admiral, who says they need to set off immediately to assist with some crisis on planet Bimbulus Prime; the captain tries to argue that repairs are still underway; the admiral sets a deadline and says ‘get it done’ and ends the call as the captain is mid-reply…
The pans around to the captain at his desk, as he puts his head in his hands, then the view pulls back and out through a window to the exterior of the ship; the title music begins to play and the titles are shown as the camera zooms around the ship showing various repairs being completed to sections of the hull, then we see the ship undock and swoosh off into space, indeed, swooshing repeatedly across the screen as the various titles are shown, then finally, we see the ship come into view of a planet and take up a visually appealing orbital stance, and the screen shows the small title ‘Bimbulus Prime’, and the action of the episode continues from there.

Of course, doing it this way would require the title sequence to be created afresh for each episode, which is obviously more effort to do than to not, but with some show formats, it’s probably feasible.

The Netflix version of A Series of Unfortunate Events did something kind of similar, but that was just a different verse or two in the theme song and some different graphics - not really what I’m looking for - so did any show ever actually do this?

Plenty of shows have done away with formal title sequences and theme songs and just display the cast credits over ongoing action. If you are insisting that the credits be accompanied by a theme song, that’s different.

I hadn’t considered that. I suppose I am looking for something that is a recognisable title sequence that would normally divide the two parts of the show

Charmed scarcely has a title sequence, but the show title always appears and is always textured and colored to immediately reflect what just happened in the cold open sequence preceding it.

The opening titles for Game of Thrones are a map flyover which changed every week to preview the locations that would be featured in the episode.

It wasn’t a big change, week to week, there were like 20 major locations in the series and each title sequence just picked out four to six, but it was an interesting wrinkle on the usual intro.

We’ve been watching the old Mission: Impossible show, and while it’s not quite what you described, they do flash clips of the upcoming episode under the title credits (with theme song).

I’ve seen some that have a short scene, then the title. Or, in the case of Seinfeld he did a comedy bit on a stage, related to the coming show.
I can’t remember when the credits shown.

Add…Sex in the City. Carrie talks sitting at her laptop about her column she’s writing while the credits roll, then the intro music.

The recent series of Perry Mason would just drop the title card into the world with fancy graphicking. And there’s theme music, I guess, but it’s just someone noodling around on a trumpet; it’s not something you can really hum along to.

Not quite what the OP asked for, but I suspect that that might not exist. Even on an animated show, it seems like a lot of effort for every ep.

The UK sitcom Are You Being Served? always played the theme over scenes from the shop floor that then naturally led into the first scene. Other ‘Croft and Lloyd’ shows probably did this too, though I’m not going to check.

The Star Trek: Lower Decks episode “A Mathematically Perfect Redemption” eschewed the usual title sequence and instead showed Peanut Hamper drifting through space to the opening credits.

I think this rule might exclude what I was thinking, but I’ll offer it anyway: in the excellent Netflix animated show Kipo and the Age of Wonderbeasts, the “title sequence” begins with a camera pan to the show’s title, which is always written somewhere in the show’s setting. Here you can see what that looks like. It’s not exactly what you’re asking for, but it’s a little bit closer.

I think they did the same thing on Some Mothers Do ’Ave ’Em.

…like everyone else, this doesn’t fit in with your criteria.

[opent spoilers for the tv show The OA]

But one of my personal favourite title sequences was for the TV show The OA. Because it didn’t happen until about fifty minutes into the show, was completely unexpected, and was the moment that the show “tipped its hand”, when it told the viewer “this isn’t what you were expecting it to be.”

The opening to Angie Tribeca would begin with a CSI-style “yeaaaaaaaaah!” scream that would segue into a cop doing something painful:

I think that’s the closest we have so far, in that it’s part of the animation of the cold open, and it’s different for each episode. The background fade to black is a point against it, but it comes close.

The Australia/New Zealand series My Life is Murder is much like you describe. It starts with the story and the title appears after the first scene, usually for about five seconds or so, as part of the background (for instance, in the topiary). Sometimes the character walk in front of it.

Similarly, I, Spy with Robert Culp and Bill Cosby, had a title sequence consisting of the credits and Culp on the top half of the screen reacting to clips from the show at the bottom, for instance wincing when his character got beat up.

One Green Acres started with Lisa picking eggs, each of which had a credit on it - which she saw and reacted to.

Didn’t ‘Police Squad!’ run a different title sequence each week? It also included a weekly ‘guest star’ - who was always killed during the opening title sequence.

Given that the title sequence is a minute or two out of a 23-46 minute show, and all the other minutes of the show have to be created afresh anyway, it should be feasible for all shows.

Similarly, Castle would always run the cold open, followed by the show logo and recognizable music tags. The show would continue from there with the credits eventually running over the action and dialog.