Gate check aka valet check requires a pink tag. Yes it is required. If tagged with normal white bag tag it can only go to bag claim. They can't give back as they have no way of knowing if it is an allowable item airside at point it is taken and loaded. At this point it is no different from any bag checked at front desk.

To clarify, they can't be sure that it was cleared for airside possession at that point. It obviously was cleared when handed over. However as not pink tagged they have no clue it was checked at gate or at front desk at a later point. Someone could hide say a knife in a part of pusher. It would be good to go as checked baggage. However it wouldn't (well shouldn't) get through TSA to airside though.