According to The Internet, it costs about $10,000 total to get a pilot's license. That means I can spend $10,000 on building a flight simulator rig instead, right?

Why just settle for a tool that allows you to live in a room, rather than go explore the real world? Simulators have their place, especially in ground-school applications to explain flight concepts to newbies and re-qualifying pilots.

Because I've always wanted to fly a plane but that seems like a terrible idea for a number of reasons

Just practice Aaron. Take a TIF(which isn't expensive) and find a good instructor that you can relate to. Doing take-off & landings needs to be done frequently at first to gain the insight, but other than that, flying can be done at a leisurely pace.

So much tech involved now with Garmin GPS screens compared to when I started. Still need to be able to read a map, a weather chart and a compass.

How did you master an ATEM mixer if it wasn't for doing the hours of practical work?