We're back. I spent much of today mentally composing my forum recap for this thread because, as you'll see (ba-BAAAAAAA...cue the ominous music), your advice proved to be very wise.
Rather than invest in single-use carts for just transporting the carseats, I opted to buy two inexpensive folding luggage carts (these). I love how fast the carts set up and fold down. We were not charged to bring them on the plane. Because they don't have latch anchors or other carseat-specific anchor points, Mr. Janine had to cipher a bit before figuring out how to strap the seats to the carts. Once he did, though, it was fast and east to set up and break down. Mental note for next time: practice this BEFORE trying to board the plane. It's also worth noting that nobody is able to roll a carseat down the aisle of an airplane. The aisles are just too narrow. You'll have to disengage from the cart, fold the cart, and carry your seat over your head down the aisle. This was unexpected and irritating; we were also wrangling kids and carrying other bags. The airline staff were only helpful when it became clear that we were holding up the works (but I'll save that for another rant outlining in great detail how much I hate Frontier Airlines).
The carts weren't easy to use, either. They are a bit awkward and the center of gravity is in a weird spot. The center of gravity is in an even weirder spot when a child rides in the carseat as you are moving the cart around. Our guys actually preferred riding in the carseat on the cart over walking on their own, which surprised me. It made the kids far easier to wrangle and definitely kept them calm, which reinforced in my mind that bringing our seats was a good choice.
We saw several families with young kids in the airport, and nobody else lugged their carseats. One mother told me she rented a carseat from their auto rental company and that they had 8 seats to choose from, all of which were in good shape. Another mother checked her son's carseat and that appeared to go all right. I didn't see anyone gate-checking carseats on this trip.
Having our own carseats also made the kids more relaxed on the plane. In fact, one of my kids fell asleep during some pretty wicked turbulence. The other kid fell asleep right after the wicked turbulence. I cannot tell you how happy I am that they were strapped in with a 5-point harness for that experience. I know it wouldn't have saved them in the event of a tragedy, but during the rough air it absolutely, without a doubt kept them contained and helped our whole family feel more secure while we were being ping-ponged around in the air.
Having our own carseats also kept me from melting down in a puddle of panic when Mr. Janine called to tell me that he and kids had been in a wreck. Nobody was hurt, but the boys were clearly excited about the event because they talked about it for hours afterwards. Knowing and trusting the carseats made all the difference in the world to me during the half-hour drive from my location to the scene of the accident.
So thanks, @Toblerhaus and @terayon and @haraya, for your advice about car seats. I am very happy with the choices I made and I'm grateful to have had your experience and opinions for guidance!