My rebuttal: Your secretary does not need to know (taken from the sheets you posted) what an angle given by a line bisecting two parallel lines is...These things all have a use, but they don't have a use to everyone...
Your idealistic answer seems to argue for changing the relevant topics. That's fine. That's an argument against the curriculum, not testing. All testing does is measure whether the curriculum is actually being taught.
You know what generally gets better test scores out of kids? More time spent studying how to answer questions and rote memorization of test topics. All done studying that stuff? If you study it more maybe one kid does better and we get more funding.
A relevant curriculum is going to be different depending on what someone's interested in and what jobs are available - and what you measure and incentivise is what you get. Testing is not politically neutral.
There isn't anything wrong with testing, in principle, at some point you have to see whether someone can do the job. But the decision as to what to test, when, how... that influences what you're going to end up with. Just standardising the tests and running them often is a bad approach. That's what I'm saying.
I'm curious though: why do you think that, absent testing, teachers would be ignoring the prescribed curriculum and secretly implementing your preferred curriculum instead? If they are not secretly doing this, then without testing teachers wouldn't be doing what you want them to do anyway.
Harmful? Really? Please tell what harmful education is coming out of standardized tests. Even the standard test taking strategy itself is useful in a general sense. You don't know the solution so narrow down to ones that seem reasonable, pick one and move on.
In my charity work I regularly interact with people whose education has been harmful to them. They failed, and from that failure gained the impression that they were stupid, and/or unable to learn specific subjects. Even putting aside the harm of taking up someone's time to no profit for themselves, you learn more than the information in the curriculum. You learn an attitude towards education, an assessment of how you relate to it. What many people learn is that it's not for them.
http://akaptur.com/blog/2015/10/10/effective-learning-strate...
I think we need more methods for teaching (I wish I had Kahn Academy when growing up!). Everyone gets to where they understand basic math and reading comprehension in their own way, and that's fine. But if we are not checking up on students progress and making sure they have some basic level of general education, we are failing the students.
It's an argument against a specific form of testing. Frequent standardised tests; testing the same things across all the schools and all the students; using that as your measuring stick, shapes the fact that you're going to get a lot of unneeded, even harmful, education. And the more frequently you run those tests, the better your 'data' is (data which is not politically neutral, as some might think) the more pronounced the effect. Test scores are the incentive.
There's two answers I can give to your question. One is the practical answer, as things stand at the moment. The other is the idealistic answer, as things could stand.
For the practical answer, the most that I can say about someone who matches that pattern persisting in education, in the current system, after grade 6 is that they may be getting something else out of some of the other subjects that they study there. If you were a parent, and you couldn't afford to educate your child yourself, then it would probably still make sense to send your child to school beyond grade 6. If you don't do so, they won't have the qualifications to get by particularly well in life, and they may learn something useful in the process (though I wouldn't bet too much on it.)
Besides, you still need some where to stick your children while you're at work. If nothing else, education is free child-care for poor people.
That's not a good argument, nor is it an argument for the current system over the potential alternatives, mind. It's just saying that sometimes crap's the best thing on the menu.
For the idealistic answer: What I'm advocating for is a different model of testing, in which the statement that someone would need an education 'beyond' grade 6 is not as meaningful. There are different skills, different areas of expertise that they could go into, but it would be very difficult to rank those skills: Is someone who we've certified can refit a central heating system more or less skilled than someone who we've certified can rewire a house? Someone with both is more skilled than someone who only has one, but at the same time are they a better electrician than someone who only has one? Is someone who understands the mathematics underlying musical theory more or less skilled than someone who understands the mathematics underlying computational complexity? If you can apply Bayes Theorem, are you more or less skilled than a frequentist statistician? It's a lot harder to rank people when you throw out the idea of 'this year,' or grades, and just have specific skills and people who can or can't do the job.
Ideally I'd be able to answer your question by saying that children are given the chance to immerse themselves in a wide range of different subjects and projects - to follow those that appeal to them as far as they can go so that when they leave they have a wealth of ability that carries them through their lives. That if you're interested in mechanics, for example, you should be able to study mechanics all the time you're at school. Wouldn't we have some fantastic mechanics if they'd been learning on cars and lorries and boats and so on every week day for five or ten years under coaching appropriate to their skill level? Wouldn't we have some fantastic builders if they'd been learning building under that setup? Fantastic artists, fantastic musicians. Especially when we could focus on those who were interested. And who knows, they might - as they get further in that area - discover that maths is useful, and then they could go and get some maths badges when they've got a passion and a reason to drive their learning, (rather than it being something forced down their throats with no real promise of a relationship to their actual interests materialising.)
But I can't - and so the former answer is what we're saddled with. Do you need it? Sure. Unless you've got reasonably well-off parents, you've got to go through the paces. Should you need it, in that form? No.