Smaller districts don’t have the overhead funding to be effective. Modern schools need technology. A district with 1000 kids doesn’t have the money to maintain competent staff or hire competent third parties.
Layers of management get introduced to keep the top from being overwhelmed, but the funding doesn't scale to account for that. As it grows, those increasing layers of management create a disconnect between the top and bottom. With the scale comes a need for larger projects, but the disconnect makes it hard to get those most out of those projects. And as the district grows to include more and more students, it becomes harder for the needs of smaller demographics within that student body to get represented.
My wife is a teacher for a large metro school district. I hear stories day after day of how her and her coworkers get handcuffed by a highly bureaucratic school district and can't get the support they need to help students succeed in the classroom. Decisions get made regarding funding, resources, and curriculum at the top from people who she never meets. These decisions are often made to cut costs to make room for growing admin needs and controversial district projects.
I work for one of the largest corporations in the US and I see plenty of bureaucracy within it, but the focus on maximizing profits at least keeps the company focused on making effective business decisions. The only thing that motivates schools is school board elections and the good will of its employees. The larger a district gets, the less power individual employees to improve things and the less representative a school board is of its constituents.
By far the most successful school districts in my state are the "medium" sized ones, with around 500 to 1000 students per grade. The districts that are much smaller or larger than that tend to have fewer resources available per student.
My sister was a teacher in a small urban school district. She had 50 kids enrolled in her 1st grade and never saw more than 20-25. Half of the parents were <25 and a ~6 were in jail for some part of the year. She would go shopping clearance sales after Christmas to buy winter coats to give to students. Some kids weren’t potty trained. Most had unstable living arrangements.
Bigger districts see both of those worlds at scale. NYC has 1M students and probably 30-40k in the conditions I described. That scale drives specialization, policy, etc.