I suspect they are just as happy to take some kids parents money for the seat as not.
And judging by the number of people I know that get upgraded, its not like they sell out every flight.
You are more likely to have an empty seat next to you in the back of the plane than in the first class.
But they don't get full price for it in any way. They'll take money to seat kids there if you wanna pay.
In the later 90s, I used to try to upgraded at check in. It was usually a 2x cost of the was a seat or less vs 3x of you got first class ahead of time. Since the 00s is pretty much never an option. Most flights are over sold and getting a decent seat (prefer front row) is really hard.
And the past couple years, even then they're often sold out when buying tickets less than months ahead of time. Mostly have traveled in occasion for work and mostly shorter notice.