I just graduated from, and my mother is a teacher at, a private school in Spain. Since you are clearly focusing on public schools in America, I'm just going to reply to parts of your argument that apply here.
They exist, in my mind, specifically to hold their members hostage from the public good to demand greater wages, job benefits, or post-retirement benefits.
If you are so worried about teachers going on strike for a few days, then you should be even more worried about people not wanting to become teachers in the future. This will happen if people view teaching as an undesirable job, due to low pay and a lack of benefits.
What is particularly disgusting to me is that many take offense to the idea that teachers should be subject to quality assurance, like evaluating them based on their subjects' grades or their attendance/participation in teacher conferences, continuing education and frequent in-person tests to make sure teachers are getting better at their job.
Even the best of teachers can have a bad year, in terms of the students they get. This doesn't mean they are suddenly bad teachers, it just means that they happened to get a group of children which generally didn't work as hard or was not as intelligent as the last year's group.
Teaching is a very busy job. A teacher can spend hours marking every night and still have more to catch up with. Attending conferences isn't always possible.
I don't know about teacher wages in America, but teachers here (at least in private schools) have neither the time nor the money to further their education while continuing to teach.
If by "in-person tests" you mean sit-ins, these are incredibly detrimental to both teachers and students. Teachers are put under pressure to plan a special lesson just for the observer, and as such will probably not deliver this lesson to the best of their ability. Students will also not want to participate for fear of being wrong in the presence of said observer.
No problem - all teachers can go on strike and the entire education platform is completely ruined for all children of a city until demands are met or a compromise is made.
This is preferable to the teachers underperforming the entire year due to a lack of motivation.
I'd love to think that all teachers are moral paragons who will work their tails off to get better at what they do and make all kids smarter,
They honestly are. During my 15 years at the same school, I have only met a handful that clearly didn't have an interest in teaching.
but clearly that isn't the case with how terribly the US education system ranks d.
Teachers are not the sole reason for bad education. This should be pretty obvious.
They could provide performance based awards to teachers of a school that together perform well, not unlike bonuses in business for departments that beat sales goals, etc.
This would be incredibly detrimental to the students. Their education would become quantitative, not qualitative. In something so (generally) subjective as education, this is a terrible idea.
If teachers were not happy with their salary they could leave and do something else instead of interrupting the education process of every child in the City.
Leaving would interrupt the education process. Changing teachers mid-year is terrible for students. And by leaving and not fighting as a union, the wages would just get worse and worse until teachers don't even want to teach there. And then you would be back at square one, paying high wages just to get people to work there.
Again, I didn't go to a public school nor do I live in America, but I'll say this - teachers genuinely do have the best intentions. Attempting to quantify education in order to make it more efficient is detrimental to both the teachers and the students. You should trust that the teachers want what is best for them, but also for education as a whole.