To be precise: Overhead does not pay salary. Overhead is only for a subset of indirect costs associated with the project as a whole (e.g., it takes place in a building, with staff and janitors and lights). PI salary must be specified as a direct cost on awards.

Second, there's theory, and there's reality. The reality of funding, even for soft-money faculty, is that schools tend to be (recognizing that not all fit the mold) a little bit flexible about how they calculate these things. Money comes in, salaries get paid, and the bean-counting happens at a slightly more fuzzy level of abstraction. The exact requirements on soft-money funding are typically negotiable, and in many cases, can be pooled across multiple researchers to create a bit of a backstop in the event of temporary funding drops. The tuition fees that your advisor's grants brought in will usually be split between the university and the departments in various (ungodly annoyingly complicated) ways, but it's fairly common that the department -- who pays the salary -- sees some fraction of indirect benefit from grad student tuition paid.

So I agree with your statement as phrased: "directly to him or his salary", but there's a ton of indirect money flow in universities.

(Source: I've had the ... mis? fortune of serving on some of the finance committees in the past, and discuss this issue regularly with colleagues at other universities, including those at medical schools.)

But as a non-me source, here's Wisconsin's definition: https://www.education.wisc.edu/soe/about/leadership/committe...

"Other types of soft money may include generated program revenues and flexible internal monies such as those funding credit outreach timetable courses." -- your tuition was generated program revenue.