The majority of noncommercial broadcasters aren't publicly funded by a tax. Most noncomms, despite being subject to the underwriting rules, don't rely on a tax for funding. It doesn't appear that the two have any connection.
Source? It seems to me that the majority of non-commercial broadcasters are owned by some sort of entity receiving tax money. I haven't looked at this nationally or even regionally, but looking at the list of non-commercial stations in Indianapolis, as provided by Radio-Locator:
Religious Organizations: WJCF, WHCN, WSPM, WQRA,
Private Universities: WICR
Community Broadcasters: WIRE, WITT
Total Non-Comms Not Receiving Tax Money: 7.
Public Universities/Schools: WJEL, WRGF, WKPW, WBDG, WEDM, WHJE, WEEM, WFIU, WBAA
Public Broadcasters: WFYI
Total Non-Comms Receiving Tax Money: 10
It is worth noting that there are at least 5 high school stations in the "receiving tax money" list. I don't believe that is typical.