How much does university hiring depend on H-1B? I would expect much of that comes through O-1 or EB-1/2/3, no?

H-1B is the default visa for international faculty hires. You can get it in a few months with relatively little effort. O-1 is more expensive, takes longer to get, and requires more effort from the applicant. Then there is the subjective approval process that involves a degree of risk, and in the end, you get a slightly inferior visa.

Green cards are almost useless for hiring, as the processing times are too long. "We would like to offer you this position, but conditionally. We still need a year or two to handle the bureaucracy, and we can't say for sure if we are actually allowed to hire you. Please don't accept another position meanwhile."

No, pretty much all professors who used to be international students or postdocs are on H1B.