As a senior resident, I can say that it can be potentially stressful but it isn't something I think about on a daily basis. To be honest, there isn't much you can do about it. In the end, all you can do is practice good medicine, stay up to date on guidelines, and see how things turn out. Main problem is, even with great management that goes by the book, you can have a poor outcome and the opposite (poor management and still having a good outcome is true).
Some patients will sue based on a bad outcome just because. We had a patient who had a bad shoulder dystocia which resulted in a permanent brachial plexus injury. The management of the dystocia went by the book but it was just a bad outcome. She was fine but her family convinced her to sue because of this.
Lo and behold, she gets pregnant again, and still saw us for the pregnancy even with a pending lawsuit. We recommended C/D, including MFM due to the potential for a repeat scenario and she refused until she was about 35 weeks when she finally changed her mind.
As far as encouraging people to go into OB GYN. I'm not actively recruiting med students when they rotate with us. Fact is, the speciality is a bit grueling. Overnight call can be physically demanding if you are in a busy practice and you could be taking this type of call in your 50s when your body may not cooperate. The lawsuit situation doesn't really come up.