Do you really think eliminating standardised tests would improve the education system?

I'd be far more okay with standardized tests if the requirement also banned homework until at least Junior High. Otherwise it's a race to the bottom to see which schools can suck up as much of the children's and families home time to help improve the schools scores.

How would you propose that students practice learning and studying on their own, outside of a classroom setting, without homework?

I don't propose that children do any of that. That's what high school is for, and 4 years of "practice" is more than enough.

Some of the reason schools start homework so early is that parents demand it. I've seen parents complain that a first grade class isn't academic enough because the teacher didn't assign enough homework.

There's an excluded middle, in that statement. De-emphasizing the importance may be an option.

Fair point. Let me rephrase the question:

Do you really think reducing standardised tests would improve the education system?

If so, what objective evidence can you provide for this belief?

If you measure it less, you'll have a weaker argument for how the education system is doing!

Your comment is precisely the problem: objective metrics will always support their proponents' arguments for more measurement, while any upsides to a lessened objective grading will be difficult to show because they'll involve subjective assessment or less assessment.

Do not confuse the investigator's convenience in accessing a truth value for the truth value itself.

That argument has been made many times and in great detail. See for instance the work of Diane Ravitch, former US Assistant Secretary of Education (https://en.m.wikipedia.org/wiki/Diane_Ravitch)

Edit: she was Assistant Secretary, not Secretary

School-funding and even whether a school closes or not is tied to these tests, so there would be some work involved in replacing that aspect of standardized testing in US schools as well.