Let me try to break this down differently.
First, there is currently no avenue right now to start a new licensed station, either full-power or low-power. New stations are only available during designated filing windows which take place approximately once every 10 years. If there was a filing window, the station's location (and if full-service, power, antenna type/directionality/etc.) must meet specific spacing and/or other protections in accordance with the FCC rules.
Now, if you are wishing to acquire an existing station, that can be done however if the station (transmitting antenna) needs to be moved to a different location, then it will require a granted construction permit and there are restrictions on how for and what channel can be used based on both availability of channels at the new location as well as specific restrictions on channels (as well as move distance).
Either way, the organization must be considered a noncommercial educational entity. This means:
(1) it can't be licensed to an individual, partnership or any kind of a for-profit organization.
(2) The licensee can be a school/educational institution, either public or private.
(3) If not licensed to a school, it has to be licensed to a state recognized nonprofit organization with an educational purpose. In the application for the new station (or assignment of the license), the organization will need to describe their educational objectives and how the radio station will be used to advance those objectives. (Private schools also have to demonstrate their incorporation status with a state, just like with non-school educational organizations.
(4) Remember, if an existing station changes hands to a different organization, the FCC must first consent to it first. This is known as an assignment of license.
The unlicensed Part 15 option for FM is extremely low power. Be advised there are many transmitters, especially sold by Chinese companies on Amazon and eBay that are illegal and cannot be used. Federal law (the PIRATE Act) calls for forfeitures/fines up to over $2 million for repeated operation. Part 15 AM does have a bit more flexibility, especially if the transmitter is on the campus of an educational institution.
Hopefully, this answers your questions.