I'm increasingly of the opinion that settlements should be outlawed in many cases. Settlements mean a case ends without a final ruling. If the parties could resolve their issues without a court, then they never should have sued. If they bring the courts into it, then the court process should be followed to the end.
Paxton made a request of the hospital that 1. He knew they would not comply with and 2. Was not legal. It was pure grandstanding. In a settlement, he escapes any penalty for his abuse of office. At the least, definitely losing the case would look worse for him than just settling.
Companies also use settlements to avoid being judged guilty of various abuses. Settlements almost always include NDAs and no admission of guilt by the company. So, some of the people harmed get a huge monetary payout, but the company can keep on hurting others.
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.