I can comment as I was a public defender for over 20 years.
Answering your questions, in order: Would you get time to thoroughly analyze and prepare your cases? It depends on the office you work for and the size of the assigned caseloads. In most offices, the answer is NO. In good offices the serious cases are assigned to teams with much lower caseloads, and newer attorneys get excellent training in analyzing cases. Still there is normally high volume so the mantra becomes "choose your battles." Not every case needs a huge workup, and sometimes you want to work the case up and the client says no. Sometimes you sit in a jury box full of in-custody men and women and advise people whether to plead guilty or fight the case based only on having read a police report. And sometimes the judge and DA get very testy if you take too long. The process only works well if you are prepared going into the courtroom, and that usually means working late or working at home or both.
Do you often get to help "good" people who have nevertheless either committed or been accused of committing a crime? Absolutely, all the time, far more often than you run into the "bad" clients. Most of your clients will have substance abuse or mental health issues, often both. Actually, almost all of them. You begin to see that most of your clients are sick rather than bad, but some of the things they do are undeniably bad. Helping people is the best part of the job.
Do you get to pick and choose cases? Not usually. Being a PD is not for he faint of heart.
Can you set your own schedule? Not in any office I have seen.
Are more public defenders needed? Good ones are always needed. There is a fairly high turnover in the first few years as people who like the idea of being a public defender find the actual practice less than satisfying. For the people who find out they do like it, it can be an extremely rewarding job, but the stress is always there. Trials are particularly challenging.
PD's tend to hang together. You get sort of an "us vs. them" mentality at times - the deck is stacked against you, not in your favor, and there are times when you please no one, not your client, not the judge, not the DA. But you soldier on, doing the best you can do for the client even if they don't appreciate it. You will be called a public pretender and asked if you have to win a certain number of cases before you can apply to be a DA. If it bothers you its probably not the job for you. At the same time some of the letters of thanks you will get become treasures, and you will know that within our legal system, your work is absolutely necessary to justice being done.
There is always the possibility of court-appointed or "conflicts" appointments in most jurisdictions. The advantage is that you can probably work part time and refuse a case or class of cases here and there (refuse too many and they will stop calling you). But it is not the same as being a public defender, because it lacks the camaraderie and isn't nearly as intense.
I've been retired for a few years already. Probably more than anything I have ever done, being a PD has shaped who I am, although I do have other interests and I FIRE'd because it was hard to pursue them with such an all-encompassing job.
I'd love to hear another public defender's take on your questions.
Bonus question: Patent bar? What the hell would a public defender know about the patent bar? Unless it was an actual bar of course.