^Well, less so in the '60s than the '80s-'00s. The majority of the "aliens" we saw in TOS were just humans with odd clothes, hairstyles, and sometimes complexions. We got full-mask monsters like the Gorn or Salt Vampire from time to time, but it was unusual to get aliens with the actors' features visible under facial prosthetics. The only ones I can think of are the Talosians, the Tellarites, and the Vians. Not counting Andorians since they had fully visible faces. And I thought of Ruk, but he just had a bald cap.
But the later shows definitely got a lot of mileage out of people like Vaughn Armstrong, Jeffrey Combs, Randy Oglesby, Scott McDonald, and Lee Arenberg by having them play multiple different aliens. They even managed to sneak in some familiar faces, like having Aron Eisenberg as a Kazon, Ethan Phillips as a human holodeck character and a Ferengi (the one in ENT, since he wasn't familiar yet when he played one in TNG), Rene Auberjonois as a Kantare, etc.
But the later shows definitely got a lot of mileage out of people like Vaughn Armstrong, Jeffrey Combs, Randy Oglesby, Scott McDonald, and Lee Arenberg by having them play multiple different aliens. They even managed to sneak in some familiar faces, like having Aron Eisenberg as a Kazon, Ethan Phillips as a human holodeck character and a Ferengi (the one in ENT, since he wasn't familiar yet when he played one in TNG), Rene Auberjonois as a Kantare, etc.