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I mean, it definitely has it (take it from someone who exclusively plays on controller). Maybe it just has trouble recognizing certain controllers, or maybe there's a problem with yours?

I think what's happening is the game has XInput support, while DS4s use DirectInput which is technically legacy/deprecated nowadays. You could run it through Steam by adding each new patch as a non-Steam game (bottom-left of Library screen) which should auto-convert inputs like it does for full Steam games, or there's third-party input mappers/wrappers out there. I don't have any dualshocks to test but my RetroBit Saturn pad can switch from XInput to DirectInput mode and the game only likes XInput (I also had to set the d-pad to left thumbstick mode for diagonal inputs, but that looks to be specific to this pad)