Like the collector in the supermarket who scans the products and takes my cash? Surely it must be impossible to automate such a complex task.

Industry/Economy lacks behind state-of-the-art technology by decades.

That's a poor example, since self-checkout scanners have existed for a while now. But notice how they aren't used exclusively. The bigger orders still require the manned scanners, and the self-checkout always has someone on duty.

A better example is plumbing. How would you go about automating a human plumber who handles all sorts of piping and crawl spaces in a large variety of settings?

Self-checkout is not automated, it is just the customer who does the work instead of an employee.

It was an example of a job which can be automated (and as you pointed out, is already automated), but which is still dominated by human workers. You suggested that all the jobs done by humans can't be automated. My response is: most (or at least some) can, but aren't.

Sure, plumbing is a much more challenging example. But it is an economic problem. The cost to automate plumbing is much higher than the utility of it.