Fake Tears, Embargo and Flatter are status moves that so many Pokemon can learn, not just this trio.
Fake Tears can be learnt by so many non-dark Pokemon, including other Psychic types such as Smoochum and Jynx. Fake Tears is notably learnt by either cute, baby or girly Pokemon, e.g. Teddiursa, Plulse, Minun, Zorua, Bonsly and Mawile. Gothita and its evolutions are also considered "girly". I mean, Bonsly can learn 3 dark moves via level up, and it's a Rock type. There's no logic there.
Embargo, besides being a useless move, can also be learnt by other Psychic types, notably Lunatone and Solrock, which I would say are weirder candidates for this move.
As for Flatter, there are only 2 Dark Pokemon that learn it by level up, being Vullaby and Mandibuzz.
Additionally, Mawile can learn 5 dark moves via levelling, and it's a pure steel type. How come?
So I don't think that a Psychic type learning dark moves is that extraordinary, considering that the dark moves are mostly status moves, and the one move that does deal damage, Faint Attack, can be learnt by so many non-dark types that it's not surprising.
You can say that reason Gothita and its evolutions can learn 4 dark type moves via level up is because they're "dark" looking Psychic types, especially noting that "goth" is in all three of their names, and the Pokemon itself is based on Goth Loli fashion.
It's also possible that the people who designed movesets just needed something more to give to Gothita and its evolutions, and decided to fill with some dark moves.