It's not rocket science. People aren't getting paid for the value of their labor, and productivity has rocketed up while wages have stagnated for decades. Your plumber charges 200 an hour and doesn't make anywhere near that much, because they work for a company who profits massively off their labor and exploits him as far as the law and supply and demand will allow.
Do most plumbers responding to house calls work for an employer? There aren't many economies of scale in that work.
Being an independent plumber means handling: marketing, networking, customer service, etc, plus plumbing. Like any other freelance position it's not easy to break in and get enough work to live. And we're not in the days of people just opening the Yellow Pages and calling a random plumber. Now it's a labyrinth of competing search engine results, yelp reviews, HomeAdvisor references, etc.
Some people will take the time to find an independent plumber. Most people call RotoRooter or whatever company has advertised the most in their area. The plumbers, electricians, handymen I call are all independent, but they're also old, at the end of their careers, having a set of regulars they could take with them going independent, and having saved a bunch of money because they were getting paid fair wages in their prime. There are economies of scale in the skilled trades, they're just not the trades themselves.
If there is no work, the plumbers don't get paid.
Also, you're probably paying the company which manages the plumbers, of which the plumbers don't directly earn that.
If you're hiring a plumber directly, I'd assume it's a more experienced one running their own business. In addition to charging you $200/hr for the work, they're probably doing their accounting and business management, billing, ordering parts and equipment, replacing tools, and communicating with clients.
I could see that only resulting in 10-20 billable hours per week.
Also, if many of their jobs are paid in cash, it's possible there's a significant amount that is unreported to the agencies generating these statistics.
For someone to come work in your house for 30 minutes, they have to compensate for 2-3 hours of time doing everything else.
Also it sounds like you live in a high cost of living area if you are paying that much for plumbers.
Sure there are a few good jobs plumbing new commercial construction that makes good money, but there is far more need and demand for people to crawl down into some residential crawlspace nobody else has been in for decades to fix and retrofit old garbage and clearing toilets and replacing wax rings or installing faucets or garbage disposals. Those aren't really money making jobs, especially when you have no idea what you might run into once you are down there.