Well, it's not their own money (to spend on coke and whores), but typically those are still their research funds so they have to make a decision as to whether to publish or buy new equipment, supplies, or get another TA/RA position. Often the funds are pre-loaded with overhead for the department so it's more expensive than it appears.

Indexed preprints make a whole lot of sense, if they'll let you do it.

Well, to some extent. If it's their own research funds, they will have to consider what's better for their career, and often, that publication will win over new equipment, for example.

That said, there are often funds available from e.g. the library, your funder, or even nationwide funds such as in the Netherlands and I think the UK, that will cover the costs of making articles available as open access. It's pretty much a no-brainer to use them for researchers.