Given the very long timelines in making an animated show, Prodigy’s storyline for the season would have been done, and in this case the dialogue was recorded, while Picard seasons three was still being developed.

We know that Kurtzman instituted monthly meetings to short out plots and use of legacy characters sometime while Prodigy was producing the first season because there were mentions in interviews about sorting out the use of Okona between Lower Decks and Prodigy.

Asking why Picard was able to go forward with a very similar storyline is reasonable. One inference is that somehow those in charge hadn’t really taken into account the overlap in audience between a live action adult targeted show like Picard and Prodigy as a family oriented offering.

What do you mean?

It’s not the first time either. Lower Decks had Riker swoop in to save the day in a finale after Picard did the same thing.

I wish this had been addressed, but I feel like it would be easy to see Fleet Formation as a response to both the PRO and LD incidents. If you have a ship that's compromised (like the Protostar or one of the Texas class ships), Fleet Formation lets you immediately deprive that ship (or any other infected ships) of autonomy by locking it into a unidirectional control from Starbase One. If it really can override anything else, that would mean that in the event of a Living Construct or whatever, Starfleet could just lock everything in Fleet Formation and the ending of PRO would never happen. But I guess they didn't anticipate a threat coming from inside Starfleet.