I guess they figure if your incarcerated you obviously can't afford bail let alone an attorney so the court then appoints one. Which is where the public defender comes into play I doubt you can get 1 for questioning
if you ask for a lawyer they have to stop questioning until you have one. I know in a lot of places the public defender's office situation is such that they basically just have to stop if you can't afford a lawyer. They can't just keep questioning on the grounds that they believe you can afford one.