> The number of graduate student admissions is directly tied to the amount of external funding.
Minor quibble: It's not merely external funding. In many sciences (math, physics, chemistry), it's common for the department to promise funding through non-research means for a number of years. In my top school, I think physics students were guaranteed TA funding for 2 years (until they pass the qualifying exams and find a professor). Math students are almost always funded as TAs (the department guaranteed 6 years).
It's mostly engineering departments that don't do this.
The engineering departments, however, admitted students regardless of funding as long as they met the bar. On the application they'd ask the candidate if they wanted the admission to be contingent on funding.
Universities like MIT that admit only if they have funding for engineering are the outliers. Usually they're private universities.