As a side note, in some rural counties, all attorney's registered in the county (represent clients there) are considered "reserve public defenders".
The names of the attorneys are kept on a duty roster. When somebody qualifies for a public defender, the next attorney on the list is assigned the case by the Court and must take the case at the standard reimbursement rate.
The only way out for the attorney is to demonstrate that you are totally unqualified for that type of case. My impression is that the attorney must be truly unqualified or else many would find ways to get out of it. If they have defended others in severe felony trials, then they can do a death penalty case.
In a trial that my high school debate club watched, a homicide defendant got lucky and drew the best defense attorney in the County who was the father of a debate club member. (Ok, big fish in a small pond, but he was still the local best). My friend then sat with us got to see her father's best efforts live and in person- to no avail as there was over whelming evidence of guilt.