Standardized testing was actually worse pre-common core, at least in Florida. It still exists... but it's not nearly as arduous as it used to be. There was a pre-common core year that a full 1/4 of the school days had some sort of standardized testing going on in the school. It was ridiculous.
I'm sure some people are associating it with other pre-existing ills in the system. (And standardized testing is still an ill that needs to be dealt with.) But the biggest complaint is that parents who had shitty education... or simply "different" education don't understand their childrens' school work. I can somewhat sympathize with that, and while I still want to call them idiots, I can see how it might be frustrating if you ARE a good parent and want to help your kids, but simply can't without relearning stuff you thought you already knew.
However common core has been proven to work, often better than what existed before. It's easier to understand from a ground-up, building block perspective a kid might have.
There WAS some pushback by teachers at the beginning, because they often had to relearn and retrain themselves, often on a subject they'd been teaching for decades, but that's largely gone now.
I'm not saying it's a perfect system. There are plenty of problems that could stand to be fixed, or at least acknowledged in a reasonable manner. But to simply throw out a system that is shown to work, without ANY sort of plan as to what you'll replace it with or how you'll pay for it or how teachers are going to deal with yet another basic reconfiguration of lesson plans after they might have had a handful of solid years to get used to these.... without considering ANY of that is irresponsible to the extreme. DeSantis is simply saying "Get rid of it and figure it out, I got no answers for you, just do it right."