I knew TNG had lots of poker, but it was only on the full watch that I saw the first poker game in its context. It was weird to see only three series regulars joined by Pulaski and for some reason O'Brien. The scene had a few nice character moments but also a solid function: to present Data as 'not like a human'. The episode is 'The Measure of a Man', which (sort of?) does some analysis on this idea.
Then we get 'The Emissary', where Worf replaces O'Brien (who I think never appears again). Why is Worf here though? There's this whole thing about being precise, proud, and unambiguous in your actions, and poker is almost entirely a game of deception. Even if Klingons in general can handle poker, Worf in particular seems unsuited to poker. They make up for this though: Worf is apparently winning on the strength of cards alone, finishes up by saying "Klingons never bluff", and then the climax of the episode is Worf bluffing some other Klingons, so it's still all functional.
Then there's 'Allegiance', with the first 'functionless' poker game. Hang on – Troi's replaced Pulaski? But she's half-Betazoid? How does that work in a game of deception? Presumably she has to not read anyone's thoughts in order to play fairly? How does she do that? Why not bring in one of the Crushers instead?
'Best of Both Worlds' gives us Wes's only game, and is only there to show Shelby's as good as Riker at most things (Riker, of course, takes this in stride and doesn't bitch about it for the whole two-parter). Dr Crusher has finally made it to the table by series 5(!!) to make sure all the women are present for some good ol' jeez-Worf-where-did-that-come-from barely-related-to-the-episode sexism. To sum up: the poker scenes are there to set up a callback or present that week's theme more than for inter-character work. Before the final episode there are at most five (usually four) regulars at the table and surprisingly few of these games seem to be fun for the people playing – two are straight up hostile (thanks Riker) and one ends in a literal punch-up. Picard's absence was never mentioned. The game is a uniquely poor choice for the two non-human leads.
So... why this construct in the finale where the leads play poker and Picard doesn't join them? He joins them for the music recitals, acts in and watches Crusher's plays, does the fencing; there was even that Dixon Hill stuff he does with others sometimes? He has an awkward breakfast with Crusher at... some cadence. And it's not like Poker was something the rest of the cast was doing together without him (they had clearly disinvited non-com O'Brien though) – it didn't bring them all together regularly. It's as though the poker scene in the finale gets across the aesthetic of a nice bookend without actually having any substance to it.
...but I'm probably overthinking the poker?
Reply from John Colagioia
One additional data-point might be relevant, the bizarre breakfast in Time Squared, where Riker fouls up scrambled eggs and blames the eggs, while Pulaski bemoans modern culture where people don't invite each other over for extravagant meals anymore, and none of it has anything to do with the rest of the episode. It functions in the same way as many of the poker games, including highlighting Worf as extra-alien, because he, unlike everybody else, likes the bad eggs.
As to the point of it all, I think that the writers really wanted to establish the characters as friends, despite the fact that they're not at all friendly when actually working. Beyond that, I've often assumed that early drafts probably had some sort of schism between Picard and Riker, explaining why so many characters have prior attachments to one of them and/or make a big show of their loyalty. Had the show run a different course (maybe if the writers strike hadn't so seriously impacted the end of the first season and beginning of the second), these scenes might have been a subplot about Riker building some sort of mutiny...but that's only speculation.
But it's also worth remembering that the show has those weird structural "spasms" that seem like they meant something important in some prior draft, but actually just lie there, almost completely inert. My favorite one comes from The Royale, where they bookend the episode talking about Fermat's Last Theorem, but it doesn't connect to anything in the episode, so it feels like they introduced it at the start of the episode to...call back to it at the end.
See also: The number of times that the characters praise the work of the writers...
Reply from @ertchin
You might know this already, but one of the main functions of the poker scenes was "we filmed the episode and it's 3 minutes short, quick, add a scene".