That means the car seat wouldn’t be checked: it would be strapped down to a seat and the baby strapped in that.
In this case, it’s very possible - even likely, because airlines like selling seats - that baby was a revenue passenger entitled to a carry-on bag of its own.
A child seat on a plane is an illusion of safety. If the plane crashes it won't help. The only reason to bring one into the cabin (and it is a good reason) is if you need one where you are going.
> Did you know that the safest place for your child on an airplane is in a government-approved child safety restraint system (CRS) or device, not on your lap? Your arms aren't capable of holding your child securely, especially during unexpected turbulence.
> The Federal Aviation Administration (FAA) strongly urges you to secure your child in a CRS or device for the duration of your flight
Checking (at bag drop) a car seat risks invisibly compromising the protection it affords. Gate checking is likely better.
I actually think it's reasonable that the woman expects more from the airport. They levy very high margin charges for landing, departure, parking, rent for vendors, and so on. Reasonable service seems expected. Perhaps not a pre-staged cart, but at least nearby phone (or number to text) to ask for one.