The problem is fundamentally NOT a very poorly trained teaching cadre… I don't believe for a minute that the teachers who schooled me to the point I qualified for Swarthmore were substantially any better trained than those today. As Carrol has regularly argued, I think, most teachers in most schools in most communities since the start of mandatory public education were only moderately well-versed in the material they taught and even less well prepared to teach anything well. The key, if you accept that schooling's primarily about reading, writing and arithmetic (with some world cultures-y social studies and some history of war and great dead white guys thrown in for good measure), is that the whole social milieu within which teachers teach has changed… from the power of the union to the work and home lives of the parents of the kids, and from the state of the school's supplies, buildings and grounds to the actual social promise of "an education."
At the same time, you are right… the problem at hand is standardized testing, NOT (necessarily) the existence of a standardized curriculum (though most common core standards suck eggs).
A