Declining enrollment and pulled funding can destroy the ability of a district to carry out the mission tasked to it by the government. This is not just hypothetical. The rural, conservative school district I used to work for (and may, many like them as pointed out in the article) and the families that rely on them do not like school vouchers, even if they otherwise support everything else that politician does. If too many students unenroll and utilize school vouchers, the whole district could collapse due to lack of funding and screw over all the remaining students.
This is compounded by the fact that vouchers often pull out more funding that the government otherwise would provide to the district for that student. This is mentioned in the article as well. e.g. the state pays the district $5k a year per student, but pulls $8k a year for a student that uses a voucher.
Also, in a lot of these rural areas there are no other good options anyway besides the school district. It’s either the district, a scammy online charter school, a scammy fly-by-night charter school, or homeschooling.