Bureaucracy makes for lousy schools, poor curricula, and some teachers that are missed as your later post illustrates. It's not the major cause of teacher shortages, however.
::snipped for brevity::
Had schools not been monopolized by the government, there would have been places for more top-notch, well-paid teachers in some schools, and fewer positions for drones who chose to go into teaching because the certificate's relatively easy to get.
You forgot one other major component: in the forties, fifties, and sixties, bright women who wanted careers were steered into teaching or nursing to the exclusion of law, medicine, engineering, or other professions. Women who were good at math taught boys who went on to design rocket ships, they didn't get the chance to build them themselves. There were "help wanted: male" and "help wanted: female" lists in the papers. Even into the seventies, a lot of professions still were not open to the brightest girls, and high school guidance counselors strongly encouraged them to go into traditionally female jobs.
They don't do that now, and most of those smart girls are avoiding teaching like the plague. Meantime, those smart girls of the fifties and sixties are retiring, or have already done so. Those left teaching, at least in my county's schools, are (barring a few undeniable flashes of brilliance) pretty much the B Team.
Then you hamper those teachers with ridiculous rules and constraints, give them a fairly needy (and frequently without home training) bunch of kids, a school board more interested in scheduling the right holidays off than in whether the kids are learning science, and it's a recipe for disaster.