From the paper you cited:
While total per pupil expenditures may be greater for the typical Black or Latinx students’ districts, there is variation in how these dollars are spent. On average, both Black and Latinx per pupil expenditures exceed White per pupil expenditures for administration, instruction, and social services. On the other hand, Black and Latinx per pupil expenditures are less than White per pupil expenditures on infrastructure. Black per pupil expenditures on all other items are slightly less than in the typical White student’s district, while Latinx spending is comparable.
Table 1 also illustrates some familiar facts. Black and Latinx students’ typical districts tend to have higher levels of child poverty, on average, than the typical White student’s district (5.4 and 4.0 percentage points higher in Black and Latinx districts, respectively).
Poor students who come from food insecure homes or live in high crime areas need more help in school in terms of school lunch programs, after school care, mental health intervention, etc. As such they need a lot more funding per student.
Complex problems require more than a simplistic solution. There are cultural, sociological, and economic forces at play. Vouchers are not a panacea. For one thing, private schools don’t have to accept students they consider a problem. Wealthy schools don’t want too many undesirables so won’t accept vouchers. Public schools will end up having mostly all the problem students without the resources to deal with them.