The part about synergy between faculties is accurate. Having strong pure math academics helps out the various applied math departments in indirect ways that I've seen.

This is a shame. Pure math is a discipline with the most out of whack discrepancy between public positive externality and private benefit, even more than in physics. Pure math discoveries often have unexpected downstream benefits decades later as they percolate slowly into applied domains in unexpected ways, and its discoverers get little credit and no financial reward. Terence Tao should be getting paid in the millions for his work.