I worked for the airlines for 13 years and I'll assure you that wheelchairs are NOT handled as car seats and other luggage is. There are laws which protect people with handicaps and their wheelchairs, by law have to be on the jetway, in good shape. The airlines get into a lot of trouble if they don't follow the law. Car seats have NO such protections. Most are just treated as luggage. The chutes that some airports use for gate-checked items are NOT used for wheelchairs, which are hand-carried down for these reasons. If the airline is not required to hand carry other items, they wont.
Sports equipment often has an extra fee and are handled differently just because of the shape and size, which prevent these items from being handled with the rest of the luggage. Again, car seats have no such limitations. They can just be thrown down chutes and thrown on trucks like any other bags.
The other obvious point which you are ignoring is what do you do if your luggage doesn't arrive when you do? How are you supposed to leave the airport? What are you supposed to use for your child? Anyone who has ever lost luggage will tell you that it can take days to get things back, if at all (usually yes but often in bad shape since it was probably sent to the wrong city).
Gate-checking is a lesser evil, for parents who don't have any other option (not allowed to use the seat on board, lap baby, etc.) Gate checked things are also forgotten, no time to load, or whatever. There's less chance of something going wrong but once it's out of your sight, anything can happen. Baggage handlers are working under tight schedules. The pressure is for them to load the plane on time. There is a reason why the airlines will NOT compensate for damaged car seats!
The OP might want to look into a Ride Safer Travel Vest, if they think they can get use with it for the future. It can't be used on the flight but would mean just one car seat to take.
The argument "I checked my car seat and it was fine" is not valid. There is no way to really know unless you're in a crash. Car seats are not indestructible and I know what checking luggage has done to enough of my suitcases! (I've flown over 40 airlines in 50 countries). I don't need my car seat going through the same abuse. I've watched baggage handlers at the airport. It's not pretty. Checking car seats is not a smart thing to do.