Part of the problem with using Psycho too much is that it gets hard to justify him surviving. Diana will snap a neck if a neck needs snapping, it's been part of her character since the relaunch. The first fight she had, she killed Decay; her second fight, she decapitated Deimos; if the assistance of the Bana hadn't made subduing her possible, she'd have killed Cheetah. Dr. Psycho is a monster to make Medusa flinch - if he gets in Diana's way too many times, if he hurts her people too much or too deep or in ways she can't fix, if he ever tries to take her on by himself, without the support of someone more powerful like Circe who can take the brunt of Diana's focus and counterattack... she's gonna kill him. "You slay the monster," after all. Which generally you want to avoid, since he's human, and we've learned that can of worms is a particularly messy one.
Why he doesn't get more exposure in the wider DCU, fighting people other than Diana, is a simpler issue. Writers just aren't that interested in Wonder Woman or her mythos, and don't think of her as having interesting bad guys. It's a shame, but on that we have at least seen some progress recently with the prominence of Psycho and Cheetah in the various Villains United stuff.