https://newscience.org/nih/ suggests that the higher indirect rates at private research institutions may occur because "universities do subsidize research out of their own pockets."
Really interesting read as an outsider. A lot of inside baseball, but my impression is that a ton of the indirect costs do fund the institution building that has happened at universities. This creates a cycle of more = more as indirect costs ratchet up and up. Surely there should be scale in indirect costs (e.g., the more money you get, the lower your indirect costs as a percentage?) In fact, I think we see the opposite in the data?