I don't see how the first study supports your point.

This preliminary analysis does not show clear patterns in the relationship between public management and countries’ performance in reading, while the level of public funding seems to be related negatively to performance only after accounting for background and institutional characteristics of schools (see Table A2.2). However, before developing any policy recommendations, further study is required to better understand the inter-relationship between public and private involvement in schools, background and institutional variables and countries’ performance in reading.

In Model 1, both the prevalence of publicly managed schools and the level of public funding for schools are not significantly related to performance. Even after including the interaction between these two variables in Model 2, no clear pattern is observed in the relationship between public and private involvement in schools and countries’ performance in reading. Similarly, in Model 3, after accounting for the student-level background variables, and in Model 5, after accounting for the country-level background variables, public involvement in managing and funding schools are not related to performance. Only after accounting for school-level background variables in Models 4, 6 and 8, do countries that provide higher levels of public funding for schools tend to have lower scores than countries that provide lower levels of public funding. The prevalence of publicly managed schools is not related to performance. Similar results can be obtained from Models 9 and 10, which account for background variables at the student, school and country levels altogether. The interaction between public funding and public management in Model 10 is not significantly related to performance.