Actually you can "sell" an LPFM, but the maximum selling price is limited to the (amortized) cost of equipment and facilities.
In other words, five years ago I got an LPFM. I paid $50,000 for studio equipment, a transmitter, antenna and related gear. If that amortizes to $20,000 today, that's the max I can sell it for. IIRC, rent does not count...so you can't charge for studio rent or tower rent...but you can count if you BUY a tower or a condo/building for studios, because those are assets that can amortize.
And your point about a paid LMA is totally valid; that's strictly prohibited. You can't pay-for-play with LPFM's. An unpaid LMA is also technically forbidden, but as I mentioned, it's a squishy sort of thing.