I never saw any "half-breeds" on the Original Star Trek other than Spock, and his very existence was scandalous. I'd say Kirk's "indiscretions" were guilt-free in that department because of alien-human DNA conflicts.
B'Elanna Torres was a human/klingon child (and had a child with a human, proving that she's not just an ass - or tigon/lyger). As was the mother of Worf's son, Alexander (who is, hinself, therefore quarter-human, the rest Klingon, even if two of the remaining quarters are culturally all-Klingon-but-raised-by-Humans)
Councillor Troi was half-human, half Betazoid.
There's been Romulan/Vulcan offspring (unsurprisingly as they're in-canon stated as virtually cousins) and Human/Romulan ones (disproving mere 'ring species' compatibility at the Vulcan/Human part of the Euler Diagram (and that
despite the problem of mixing cupro-blooded Vulcan and haemo-blooded Human traits… somehow! You aew right to be sceptical!)
I recall there being a Bajoran/other hybrid in DS9, but can't remember the full details off-hand.
And all this is 'explained' in the TNG episode where it is discovered that many (all?) different galactic races are linked by DNA 'clues' left by some progenitor species of some kind, who seeded each races biogenesis origins with fragments that 'do stuff' when combined, even after aeons of evolution.
(How this meshes with Q's showing of Picard the moment that primordial slime started life on Earth, I don't know. But I'd probably trust a miraculously self-rebuilding chemical 'program' over that trickster Q any day, so I'm giving the DNA message the benefit of the doubt.)
Although Gene's episodes DO often seem like psychotherapy sessions, each character of the Enterprise a one dimensional reflection of his own psych -Kirk is the ego- naturally he can't help being a jerk, he's not a whole human being just a projection of primitive instincts, Spock the intellect, etc, etc.
In TNG, they were slightly less 1D. Even ignoring pre-/post-Locutus Picard (or pre-/post-emotion Data) they generally mixed and matched the singular qualities of the TOC crew into newly rearranged packages for the TNG one. Data takes logic (and strength) from Spock, but the majority token-Alienness goes to Worf (also strength, but note the Worf Effect, which was basically the Tasha Yar Effect made plot-proofly non-lethal) and the mental
alienness sprinkles onto Troi (who otherwise takes on Ensign Rand's token feminine dimension, given Yar never used it when not affected by Random Space Phenomena, Crusher already had her kid to deal with at first (Pulaski never strayed far from the professional), the females in engineering before Geordi/during his time were basically techheads, etc, etc).
Data vs Wesley vs Geordi is certainly very Rock/Paper/Scissors (or 'one goes rogue, the other two might need to team up to handle the other, two
of them rogue at the same time needs Picard or Riker to trump the duo in a psychological manner) in a way that nobody
could best Scotty if he was on proper form. Except for a GNDN pipe at head height in a Jeffries Tube.
Also, I have no idea how old Picard was supposed to be, but in a population of futuristic humans that get to be a couple hundred years old, in addition to the real-world low birth rates of economically prosperous populations, humans of the age range 1-16 would probably be quite a rarity. The danger would be that they would be incredibly smothered and spoiled "-oh look hon, it's a CHILD!- Let's go see it. Do we have anything like candy or a toy or anything?"
I don't think the post-scarcity Feceration was post-reproduction in any (*ahem*) conceivable
way. We see more kids than in Kirk's time because Picard is on a 'keep the home fires burning' type of ship, keeping an ambasadoreal overview much of the time, compared to Kirk's "Where No Man Has Gone Before" seemingly perpetual First Contacting (And To Hell With The Prime Directive!)… Maybe in some way we can say that Picard carries a cresh because many more of his staff are their mothers (and/or fathers) taking their kids along. It's arguable about whether it's wise, but it's definitely more socially forward-thinking. Very '90s. Or 2360s.
That said, the age-balance of characters does seem to more reflect Central Casting's output of the time (when not trying to fulfill "loads of children in this scene"), and it's overwhelmingly mature people seen around Ben Sisko's father's restaurant on Earth (time of day/cultural drift?). Incidentally, Joseph Sisko had several children other than Ben (one on-screen grandchild) and had medical support but wasn't even anywhere near being a centarian and no indication that he'd have a living bicentennial. There were very few human Methuselahs outside of Khan's tribe (even discounting their cryogenic sleep) and Scotty (cheated with a transporter-loop). Data's 'father'? Obviously Spock (being Vulcan, still going strong for some time around a rebooting time-loop) and some notable Klingons/others, but Humans are somewhat to them as the Ocampa are to us. We just pack in a lot more depth and breadth.
(IIRC, Picard was late-50s+, a career officer, and had no children of his own. But he grew up (in a rural setting) with a brother, and had a nephew, so may not have been overwhelmed by schoolfriends/etc, pre-Academy, probably knew the kids of families with neighbouring vineyards. Though he was shy, in his early years, and gave up his musical performing; maybe why he later took to commanding like he did, with holodeck performance mainly for his own enjoyment, as a way of overcoming that.)