The FAA recommendation for carseat use on aircraft is based on weight, not age or height. A child who is over 40 lbs. will be safest using just the aircraft seat, with the lap belt.
A backless booster may be carried onto the plane as carryon luggage *if* it fits into the overhead bin or under the seat in front of you--it may NOT be actually used for seating on the aircraft. All carseats that are being used for seating *must* have an integral upper-body harness (and if sold in the US, will have a mfr. sticker stating their suitability for aircraft use.)
I always gate-check carseats of any kind if they will not be used on the aircraft. Not only are they less likely to end up under a mountain of heavy luggage; but gate-checked baggage can be retrieved in the event that a connecting flight is missed or cancelled, which is not always the case with regular checked baggage. You don't want to take the risk of having a carseat sent onto the wrong flight, as you need it to leave the airport once you reach your destination.