And thus airports are pressured by airlines to minimize costs. In the end, the answer is probably, that it's just cheaper to use windows than to invest any more effort into the task.

Windows is dominant for UIs because of a long history of having drivers for all the various peripherals which eases hw procurement issues. Sure that advantage went away long ago but the momentum takes a long time to die out. The brains with the expertise know how to do it on Windows. Combine that with the fact that Windows UI is not something you have to train new hires on and it makes business sense that all the user-facing interfaces ended up Windows applications. Sure, over time they should be migrated to containerized web UIs, but there's a lot of work to be done. Updating the flight arrival display isn't mission critical; it's at the end of the backlog.

I believe I heard one thing that got screwed up was a crew-scheduler product. Like: they couldn't check in crews or get their schedules written in so a system (presumably something on a mainframe) could work out where all the people needed to be at what times. Much more mission critical! The central big brain was (probably) not Windows, but those user inputs at the end were enough to screw up the whole business process.

Of course your operating system vendor will always have access to kernel and increased permissions to do things that other software won’t have.