There is one: a free market. Private schools don't have the luxury of doing studies forever; if their teaching is not effective, they go out of business. The ones who stay in business have a direct, quantitative measure of the effectiveness of their teachers: profit.
Parents don't have to "believe" teachers are effective; they can see the results for themselves and judge them. My parents had no problem telling which of my teachers were actually teaching me things and which were just boring me to death in class.
As for "rigorous", the whole point of the comment I responded to was that there are no "rigorous" third-party measures of teacher effectiveness. That's the "problem" that all these experts have been struggling unsuccessfully to solve for decades. The reason they can't solve it is that as they have conceived the "problem", it's not solvable. The people in a position to judge the effectiveness of teachers are the people who directly observe the impact of their teaching. Third-party "experts" can't do that.
Not all parents are that involved, and a kid being bored in class isn't necessarily the teacher's fault.
Then that's the root problem that needs to be fixed, and trying to invent ways for third party "experts" to "measure teacher effectiveness" won't help.
> a kid being bored in class isn't necessarily the teacher's fault
It might not be that individual teacher's fault, true; the kid might just be in the wrong class. But it still means there's a teaching problem.
Without that, we fall back on more qualitative, holistic methods they are both game-able and subjective.
We may see a general cultural shift towards higher teacher pay (which I believe would be positive if paired with higher professional standards), but it’s difficult to argue in our current culture that teachers “deserve” more pay when we can’t even reliably determine which teachers are having an above average educational impact.