The type of visa status that a Program will take is the decision of the institution. The Program (director, coordinator, etc.) has no choice or weight on this. If you have a Permanent Green Card or are a U.S. Citizen, you’re golden. Refugee status is usually accepted. If you need a visa, J-1s are the most commonly supported type. H1-Bs are less common these days and institutional support for them are increasingly uncommon—especially if you are not a top candidate in your specialty of choice. Any other visa type will nearly always exclude you from candidacy.
If you need a visa, be willing to accept a J-1 or apply to every single program which will sponsor the visa for which you are looking. Information about sponsorship is usually widely available on Program websites. I would 100% call the Programs to confirm visa sponsorship types if it is not listed on their website.