Go private if you can afford it!
The details you have give are a bit confusing and it does not quite make sense to charge you with being an "accessory after the fact" if the police claim that you allegedly "transported" the co-defendant from the scene of the crime, which I suppose is a whole lot better than what sounds like should be a straight forward charge of accessory to the entire [whatever] crime that was allegedly committed. But I digress.
Anyways, I am not sure why you were denied a public defender, but I can tell you that it is a legal impossibility to deny you defense counsel, unless it is shown that you are financially well off to be able to afford a private attorney, and, there is no need or place to appeal the judge's say-so, because you can very well just contact or go in person to the office of the Public Defender in your locale and ask that you be assigned one.
But if the charge is going to be such that you have explained and if you can truly afford the outlay, then, it would be a lot wiser to retain private counsel. Public defender offices and their attorneys the country over are overworked, underpaid, and understaffed and as such are not given to taking matters the long route and are more inclined to plead you out and clear their dockets, whereas a private attorney will go to bat for you proper and might have a trick or two up his or her sleeve that may work to your benefit.
And save for traffic offenses and jaywalking and such, when was the last time the police actually witnessed a serious crime taking place, which makes an eye-witness testimony a quite powerful tool for the prosecution.
fredrikklaw