Gate-checking is much safer than checking your seat in as baggage.
Many people claim they've checked seats and they've been "fine" but the truth is that your seat could be damaged and you wouldn't be able to tell. The only way to know is if there was an accident and let's hope there isn't. This is not a "purist" view but a practical one. A car seat is an important safety item. Checking it means it could be mishandled. Car seats are only meant to sustain one accident after all.
No car seat bag will protect your seat from damage or lost. The only purpose in using one is to keep the seat clean.
The other big risk is that your seat could get lost. It usually takes about two days to get lost bags returned to you. Then how would you transport your child?
The best way is if you take the seat to the gate. I fly alone with three children and honestly, taking a car seat is not a big deal. I have very active children and I've never had any problems with any "running off" and getting hurt (silly logic IMHO). My husband is never with me to help and gate-passes aren't possible.
I strap my car seat on to a small foldable metal luggage cart. I use an extra bungee cord and this is less expensive and works better than a Gogo Kidz. Your child would be safer and more comfortable in the seat than in the CARES harness (which can only be used for a short time anyway).
Then you would be sure your car seat is not lost or mishandled.