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

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."