When you're on Spectator Mode, Every monster becomes invisible and you can't see with their eyes.

Not being able to attack monsters in spectator mode is intended, getting invisible isn't intended though, did you mean invincible?

Yeah.

To see in their POV, left-click on said mob.

POV? Huh.

Seeing through their eyes (as you call it) = the entitiy's POV (Point of view)

Are you sure the monsters didn't just despawn?

Yeah, I'm 100% sure.

Fixed summary according to description (Invincible -> Invisible). Thus duplicate of MC-46776