Scale for scale's sake isn't necessarily ideal, or efficient.
My freshman year of highschool, I was at a very large school, around 3400 in attendance. The school had been expanded several times, and was more or less at capacity. Some facilities were scaled out well, others weren't. There were fights on a weekly basis, tons of security trying to prevent that, but, they could only do so much. Lunch lines were very long, sometimes I'd be lucky to get 10 minutes to eat, mostly brought my own lunch because of that. The school was locked down a few times that year because of weapon scares. It was hell.
Then my sophomore year, they cut the school in half, sending most of the students, save for the seniors, to a new school across town. That landed the school I attended at around 1400 kids, my junior year, all of the extra seniors had graduated, so, it shrank further to around 1300. From sophomore year on, there was not a single fight, everyone knew eachother and got along, they cut the security team down to just two people and they both were well liked. Test scores were way up, class sizes went from 40+ to consistently under 30, and things overall were fairly good as far as highschool goes. The new school across town was a similar situation from what I gathered from friends who went there.
Now you might write that off as it was simply too crowded, but, they closed about a third of the school, which was held in portable classrooms when the split happened. Some of those were re-purposed for offices and whatnot, most got hauled away. So the actual density of people didn't really change much. We got some extra space to have PE classes in, but that was about it, space wise.
You might also write that off as demographic changes, which was also not true. In fact, the roughest part of town all went to the school I attended, while the nicer, more affluent side of town, mostly got sent to the new school.
The way I see it, the school was simply too large, and managing that many students day to day, and all of their needs and affairs, did not scale well, and bred inefficiency, inefficiency that existed long before I got there and everyone got to keep their job and maintain that as the status-quo. Sure the buildings/campus could physically contain them well enough, yes they were all reasonably well fed and watered, but they'd become unmanageable in such a way that was not likely to be fixed.
In terms of reinvention in public education, smaller schools are a decent model, as are smaller class sizes. The real deciding factor in how well a student does however, is parent buy-in, which homeschool models have in spades, and in addition the class sizes are about as small as you can reasonably go as well. Understand of course that, parents opting to homeschool are paying taxes to public schools, while receiving no benefit, and that in addition to paying for the costs associated with homeschooling. Not all families can afford to effectively pay for schooling twice, hell, most can barely afford to exist while paying once. You could very likely fix your concerns about privilege by simply redirecting money back to homeschool parents that would normally be spent on public school. There are very well defined $/student numbers out there, so, the amounts would be fairly trivial to come up with. You could also really work around the whole right/left politically polarized bullshit factory by perhaps giving a little more to the lower income families and a little less to high income families. I think that'd sufficiently frame the nominally 'right wing' framed homeschool ideals as neutral, or at least dissonant enough that you might get most of everyone on-board provided you could get the right palms greased in state government.
Understand of course that, the public education system is an enormous apparatus, and they're managing unfathomable numbers of students. They're also not setup in fairly clean competency hierarchies either. At the local level, seniority tends to be king, at the broader level, the ability to navigate the political system is the selection factor. Neither of those have anything to do with "who's talented at educating kids", and while you do have means to measure performance, those same means are also decided upon by the very same people being effectively measured, so there's an incentive to make them easier over time, as that looks politically better. So if anything, the incentives we've created are exactly backwards, and the system cannot change until those incentives are fixed. I don't really know how you'd prevent standardized testing from being watered down or trifled with. I don't really know how you'd create sufficient educator turnover with the unions in place to weed out bad teachers. More or less, the system is broken, we need bloodsports to fix it, but they've thought of that, and have prevented it from happening.
I'm not holding by breath of course. People are more than likely going to write it off as "privilege" regardless of the changes proposed, with a whole list of valid enough sounding reasons that they haven't really thought through, most of which won't really have any baring on reality, and nothing will change aside from actual children being mortally wounded by poor education. That'll continue as-is, furthering the class divide, increasing unrest, increasing predatory behavior toward people who, could have been saved by better policy, but were thought more useful as sacrifices on the altar of "the other team's" apparatus.
As for me? I'll be eating beans and rice and driving a 15 year old honda while putting my future children through a homeschool program regardless of what happens.