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?

There’s ongoing costs, too. You can’t just /keep/ a pilot’s license. You have to use it. If you don’t fly a minimum number of hours a year, you lose it. So you’ll have to rent planes. Lots of pilots sell “at cost” trips to travelers to fund their hours.