If patents were public, could drug recipes be used by the wrong people and sold on the black market?

No, that wouldn’t happen. In fact this risk is higher today with a monopoly patent system. To see why, consider that unreliable black market production happens only when a) people struggle to get access because of price b) illicit alternatives can be provided much cheaper c) those alternatives have credibility. However, in the fund system it will be next to impossible to provide illicit alternatives that are cheaper and credible because under the fund drug prices will be low, and close to the cost of production with top quality manufacturers competing to supply at these prices. Without the inflated prices of a patent monopoly there will be no opportunity for illicit black market supply.