If the parties could resolve their issues without a court, then they never should have sued.
Your assumption is that all parties act in good faith. Every case can potentially be resolved out of court. However, Ken Paxton wasn't going to back down just because it was the decent thing to do—only under the threat of losing money and political points (and possibly facing sanctions) in a case he was doomed to lose did he back down.
The purpose of a court case or a settlement is to receive remedy for injustice. If the parties agree out of court what they each deem to be "fair compensation" for those acts of injustice, the court proceeding is rendered moot.
I agree that this was overreach, and Ken Paxton is a fucking ghoul who deserves nothing less than 30 years in a federal prison, but nobody has a legal obligation or compulsion to press charges or continue a lawsuit.
Additionally, Republicans have made a sport out of operating in those gray areas of our legal system to push their stupid culture war bullshit and maintain their thin cloak of plausible deniability. He is going to get away with it, because what he did isn't explicitly illegal, and the hospitals exercised their right not to pursue further legal action.