This is directly contradicted by
https://www.justice.gov/crt/title-vi-legal-manual#Direct :
> Federal financial assistance may be received directly or indirectly.(11) For example, colleges indirectly receive Federal financial assistance when they accept students who pay, in part, with Federal financial aid directly distributed to the students. Grove City College v. Bell, 465 U.S. 555, 564 (1984)
Here's a summary from wikipedia ( https://en.wikipedia.org/wiki/Title_IX ) of Grove City:
> The Court decided that since Grove City College was only receiving federal funding through the grant program, only that program had to be in compliance. The ruling was a major victory for those opposed to Title IX, as it made many institutions' sports programs outside of the rule of Title IX and, thus, reduced the scope of Title IX.[3] The ruling, however, was short-lived. The Civil Rights Restoration Act was passed in 1988 which extended Title IX coverage to all programs of any educational institution that receives any federal assistance, both direct and indirect.
(emphasis mine)
So, you're just plain wrong as to the state of the law; if the government gives somebody a big pile of cash, and they use it to pay for college, all federal regulations apply to the school. It is not necessary for the school to receive, distribute, or know about the federal money.