In Gen 3, extrasensory was an odd signature levelup move for Nuzleaf to have. Why does a part Dark type pokemon get a Psychic signature move? It learns no other Psychic damaging moves. And why does it have 30 pp as an 80 base power move?
Nuzleaf Signature Move: Only thing I can think of is it having to do with its
Tengu inspiration; there are a few tales where the tengu shows off general supernatural powers. Tengu are also tricksters, and with Nuzleaf and Shiftry being Grass/Dark-types, two Types they're vulnerable to is Poison & Fighting, so it may have also been given Extrasensory (which is called "Supernatural Power" in Japan) to give it a leg up against those Types.
Also, something I just discovered, the Japanese spelling of "Supernatural Power" is "じんつうりき" translated as "Jintsūriki". Looking it on Google leads to a Wikipedia definition page and a technique used in Naruto. However, changing it to "Jintsuriki" leads to a page for a
music theme of a Dead Or Alive character:
Bankotsubo, who is a Tengu (a name which he also goes by). He first appeared in the series in the 2nd game which released in 2000, two years before Gen III was released. Could be coincidence, or maybe there's a Dead or Alive fan working at GF at the time and thought, since they were making a tengu Pokemon, was able to get Nuzleaf having this move as a subtle reference.
80 Power & 30 PP: Maybe to make up for it being 10 Power weaker than Psychic? "Sorry, Psychic is still stronger, but to make up for it you can use Extrasensory more".
Also, not a lot of Psychics learn it, its usually reserved for Pokemon with yokai or similair supernatural creatures, so maybe it's also to make up for all those non-Psychics that get it who don't get STAB from it on top of it being 10 points weaker than Psychic. And for the few Psychics who learn it, just a nice bonus.
The crying out for these moves is less "this will make the Pokemon Good (TM)" and more "this pokemon has nothing and would like something so it is more usable". It was more an underline of how bad it was and a want for the bare minimum.
I don't think its even that. I think it's more the line of thinking of "this move sounds like something this Pokemon would do, why doesn't it learn it?". It's like how Lickitung didn't learn Lick in Gen I or how Sneasel still can't learn Razor Wind; these moves didn't/won't make the Pokemon any better, but it's still something they would do/related to them so "logically" should get it.
In Gen I, Let's Go and Gen VIII, Pay Day is a TM, and it's distribution seems really random.
(...)
Of course a lot of TMs in Gen I were just given to a bunch of random mons, but that doesn't seem to be the case here. It feels like there's a reason these specific Pokémon were chosen, but I fail to see what it is.
Looking through the list and seeing Psyduck gets it, I'm wondering if the majority of the Gen I Pokemon that got it was because these Pokemon were several staffers favorite Pokemon: Pikachu (which was popular among many staff members), Psyduck (which is notably Masuda's), Mankey, Rapidash, Fearow, Slowpoke, Seel, Rhydon (aka the first Pokemon created), Eevee (it having become a second mascot in recent years and especially for Let's Go), & Snorlax (which is based on Kōji Nishino). Mankey, Slowpoke, and Seel in particular feel like quirky picks.
Nidoqueen, Nidoking, & now Calyrex feel like they get it because their name/theme hint toward royalty and one thing royalty is known for is being rich. Related could be
R_N's explanation for Indeedee getting it, as royalty and plenty of butlers & maids working for them and in charge of doing errands (which usually involves buying stuff).
Mewtwo might get it simply because Mew does. Like, Mew can learn every TM, though Mewtwo can't, and usually the reason is that its a Move that Mewtwo can't perform or may not have use for. Pay Day, at least in Red & Green/Blue, was seen as possible and gimmicky enough it could learn it. So why did it lose it in Yellow? Possibly because by that time the first movie had been released and, well, upon watching the super serious & deadly Mewtwo on the big screen its hard to imagine it doing any quirky moves like tossing coins at the opponent.
And then Gen VIII gave it to a whole batch of cat Pokemon for the obvious reason of being called "Coin for a Cat" in Japan. BTW Hoppip would be included in that, hence it getting it as an Egg Move in Gen II.
Only other one is Delibird getting it as an Event Move, and hey what else do you give someone as a present when you can't think of anything: money!