What is supported is that those who have a position of power, authority, wealth, privilege are able to convey such to their children. In the US, that's been white people until very recently when affirmative action and other programs increased access to education and good jobs for women, people of color, etc.
No, those at the top levels of wealth, power, and privilege, in the US are and have been a small subset of people, who are disproportionately white, but most white people are not included among them. Whatever trait you label the privilege with implies that the trait is largely a neccessary and a sufficient criteria for possessing the privilege and that people who share the trait share the same privilege. This is absolutely false, so the term is objectively inaccurate in reference to benefits tied to wealth, etc. The benefits in Bronzeage's analogy and those shown in the OP are of this sort that do not qualify as "white privilege". In fact, many of the differences in the OP study aren't about what the white kids have, but more about the bad things that they don't have as much of around them, namely parents, family, friends, and neighbors engaged in criminal activity. The benefit of not being around as many criminals can only be a "white" thing, if you assume that being a criminal is a "black" thing.
Only benefits that flow consistently and directly from one's race could be reasonably labeled "white privilege". For example, if Bronzeage also had a father in architecture who taught him the same skills and he was equally qualified and motivated as his classmate, but 7 of 10 potential employers preferred his classmate solely and directly because he was white while Bronzeage was black, then that would be "white privilege". Such privileges may and likely do exist, but neither the OP study nor most of what people who use the term refer to are actually about white privilege, but rather are about variable advantages imparted in subtle ways that vary as much or more within a race as between races and thus are modestly but not reliably related to race. Those indirect and subtle pathways by which the wealth, power, and education of family and community give advantages to a person are important. But ironically their consideration does more to weaken claims of "white privilege" than to support them, as with the OP. The OP tries to compare "poor" whites and "poor" blacks in order to claim that any better outcomes for whites is not wealth but race, and thus "white privilege". But these subtle ways in which "wealth" exists and transfers from one generation to the next show that the OP study is not comparing groups that are similar in all but race, but groups that differ notably in subtle forms of "wealth" and resources, therefore no differences in outcome are attributable to race influence in the kid's lives but rather to wealth differences they were born into. Were those wealth differences partly or even entirely the result of many indirect historical influences of race and racism before the kids in question were even born? Yes, but that is a different question and a different type of explanation.