Only about one in three adults in the US has a bachelor's or higher. That alone means you've reduced your hiring pool by 66 percent, before we even talk about an extra year of teaching certification.
As you said, and I agree, getting a bachelor's isn't *hard.* You don't have to be smart, you just need time and money. Therefore, by restricting a job to people who have passed that barrier, you are not selecting for smart people, but people with time and money (or willingness to accrue substantial debt.)
You are correct, Stoneheart, that a lot of people who stick with teaching for the long haul are just lazy. Here comes the second part of what I was saying, which has gotten left aside-- do you know how to weed out indolence? And perverts, for that matter? You have to fire them, because they can't be fixed.
So the current problem with the recruitment and retention of teachers in US public schools is this: there are substantial barriers to entry which do not select for better employees, and once they have their jobs, teachers are hard to fire, partly because of unions and partly because the pipeline for recruitment is long and has little attraction for most people.
If you want non-rapey teachers who are willing to do their jobs, and capable of doing so efficiently, you need more incentive to be a teacher (for example, higher pay) and a higher turnover of personnel. If you have more people trying to be teachers, you can afford to shitcan more.
It sounds much easier on paper, of course. Guilds and bureaucracies resist such measures quite effectively, and US culture doesn't help in this case at all.