This makes me sad...
Teachers for the most part work in a controlled environment
They typically have a classroom
True, but while cubicles and factory workstations are usually interchangeable, classrooms are often shared, and not suited for the subjects they are being used for. Physics labs being taught in English classrooms? English classes being taught in the auditorium? This happens quite often.
equipment provided by the school
The school only provides a minimum of necessary equipment. Most teachers use a considerable percentage of their salary just buying pens and paper out of their own pockets so their students can do their assignments.
and students who are required to be there
And who don't show up...
All are internal entities aside from the occasional intrusion by a parent, bureaucrat, or local newspaper
As long as by 'occasional' you mean 'almost every day', then sure.
In comparison, plumbers almost never work in a controlled environment. They go to someone's house or office and deal with whatever is there
The world of plumbing is one of the most over-standardized industries in the world. Plumbers almost always work in highly controlled environments, because municipalities tend to have very strict building codes for what materials and construction practices are allowed to be used. The types of things that can go wrong in plumbing are always fixable by a reasonably skilled plumber - as long as the money is there to perform the necessary repairs. In fact, the world of plumbing is so standardized now that the biggest problems plumbers have is when working on fixtures that are so old the parts can't be purchased at the local Home Depot.
Teachers rarely have access to the sort of regular problems that plumbers are faced with.
They don't get to take the plumbing to a controlled place and work on it there
And teachers do? Plumbers aren't likely to have to fix 30+ pipes that are all interconnected in mysterious ways, where some pipes are running slow because they're too busy getting high to let water flow through them, and other pipes are empty because their parent pipes don't feel like it's their responsibility to keep them pressurized.
My take is that scrutiny is not that difficult, especially given that the primary goal of public school education is the education of students in a limited group of subjects. You have measures such as student performance on standardized tests, discipline actions, and the future success of past students.
Scrutiny is incredibly difficult. My company still thinks that LOC is a good metric for measuring developer skills.
And why on earth would you say that the selection of subjects is 'limited'? You think that Math, History, Language, Science, Literature, Government, Art, and all the other subjects taught in public school is limited? I would call the pretty damned comprehensive...
There are dozens if not hundreds of different heuristics that make up whether a teacher is performing well. The problem with optimization is that once you select a function to optimize, it will do that no matter what your actual goal is. Once you 'decide' that a handful of standardized test scores are what we will use to optimize, then you will see those test scores rise and rise and rise to some mysterious limit - and they will still have no correlation to how 'good' a teacher really is.
Here's a counterargument that I don't expect you to give a crap about, but I'm going to include it because I think it's important:
There will always be a distribution of teachers in the workforce. A small handful of really bad ones, a small handful of really good ones, and a vast majority of decent ones.
There will also always be a distribution of students in the classroom. A small handful of really bad ones, a small handful of really good ones, and a vast majority of decent ones.
How would you decide who is a bad student? Would you use standardized test scores?
Now can you think of a metric that measures teacher performance based on standardized test scores that can be normalized to their students' performance on standardized test scores?
It's no more fair to measure a teacher's performance without evaluating their students than it is to measure a student's performance without evaluating their teachers.