It seems that Psi operatives are trained outside the combat field, and the best tactic is to simply keep them training in the psi lab, so they don't get injured and reach your last missions with all the psi abilities unlocked. Does this mean that a fresh psi operative is equally good as one who has fought 20 missions and killed many more aliens or do they also get stat improvements through alien kills like regular soldiers?
Psi Operatives do not receive stat improvements via enemy kills, but only through training in the Psi Lab as you know.
If you have War of the Chosen, there is a small benefit to sending Psi Operatives into battle. Like your other soldiers, they can gain Ability Points if they are sent on missions, which are added to XCOM's shared pool. Additionally, it may be necessary to send Psi Operatives into the field to avoid sending Tired soldiers on consecutive missions.