what makes a d&d sub race stick?

[KorvinStarmast]

The three most common themes I encounter are (in no particular order), contemporary media emulation, special snowflakes, and horny.

That's a pretty good summary, I think. To add to that, people doing a copy cat of Matt Mercers play group. Seen a bit of that. CR has had an impact.

I’ve found the lack of lore is the biggest flaw with the animal sorts. Without common ground within or outside D&D they’re functionally little more than a costume.

Yes, there's more but I won't go there. My own dislike for D&D animal races is based in getting sick of the cat people in space trope during the 80's. (I read a lot of SF while at sea, to include various FR novels). CJ Cherryh had these cat people in a bunch of books and it just turned me off. When tabaxi showed up, I had an averse reaction.
The aarkocra is almost palatable.
The Kenku are awful. (At least in D&D 5e). It might have been an interesting idea at one point, but the implementation was a failure.

Tieflings benefit greatly from being tied into lore. I may not like what has been done with some particulars recently, but there’s no confusion over where they fit into a setting.

Yes, and I seem to recall that they fit into Planescape pretty well. I prefer Tieflings without the aesthetical mess that is the current horns and tail nonsense. (I guess we can blame 4e for that?). Having a bloodline to a fiend or a demon can be far more subtle and still be quite effective, in terms of lore/world building.
A problem with elves is that Tolkien had his taxonomy of elves and I think that led to the 'sub race' of elves thing. (Can't prove it, but that's my suspicion).

[And by the way, your guess missed by a whole generation. My dad didn't have a favorite rock band. He had a favorite swing band, from the forties. I have a favorite 60s rock band.]

My dad's fave is John Coltrane. Jazz is Mom and Dad's thing. (They went to college in the late 40's / early 50's.)

One more fact: just because your group doesn't play them, you can't conclude that nobody does.

True. My days of playing halflings are long gone. (I kind of over did it in OD&D and AD&D 1e). I'll still play a dwarf here and there (have a celestial warlock dwarf at present)

the anthro options have baggage and are hard to work with both from what the animal is like and other media which feature anthro animal people.

Yes.

I do not think about people who grew up in the 40's as my only relatives from back then are no longer with us.

My dad was born before the great depression started. He's still alive. Then again, he never played D&D, but he didn't stop us from playing it.

I want to know what seems to make something stick as a second part, why is dwarf seemingly condemned to be nothing but clones?

Lack of imagination on the part of players, perhaps?