I'm not sure what source you're using, but no, it isn't even double the median US salary ($56k): https://www.bls.gov/oes/current/oes_nat.htm

It's good money, don't get me wrong, but it's not immediately worth it if you spend $200k paying for law school.

Edit: I misread the data above, but I'm finding similar numbers from other sources, so I guess it depends on where you look. Regardless, you're correct in your overall point, I just wanted to point out that a $100k salary does not necessarily make it worth going to law school.

With that said, you're right: if your goal is to go into law to make money, it shouldn't be very difficult earn well into the six figures a few years in. I imagine most competent lawyers earning less than that are working in public defense or a similarly underpaid aspect of law, knowing the tradeoff. For reference, the median salary for software engineers is only $110k, but that is clearly on the low end of what a competent engineer can make after even just 5 years in the industry.