It would be a nice thing to do from several angles I think and would help drive interest in radio as a technology in general; as well as building transmitters and aerial systems, there's things like setting up an audio processing chain to get the best possible modulation and being a disc jockey which will always have a bit of a buzz from being on the 'real' radio. If valves and vintage equipment are involved it would be an interesting form of technology history preservation and 'living history', and there's also an environmental angle where old analogue radio receivers could be prevented from becoming e-waste.
There could be a minimal licencing regime to demonstrate the individual is not a complete numpty and a fee to PRS for music rights along the lines of how streaming works. Maximum power could be kept low especially at night to avoid interfering with countries in Europe where AM radio is still an active platform, and I don't think there'd need to be a lot of enforcement with respect to content since you'd only realistically be broadcasting to other anoraks. Additionally there's already precedent for non-profit stations where the medium of AM itself plays a role, for example former pirate Radio Caroline which uses it to keep its historic radio ship in operation.