Isn't that what the article is saying? Less research funding == Fewer admissions.
> 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.
While it is true that departments often fund their graduate students for the first year (or possibly 2) out of their own budgets, their budgets are largely determined by the generosity of their Deans (who got the money from indirect costs from grants) or their own indirect costs. And they will not be admitting students if they do not see a clear path for them to be externally funded after their first year or two.