IMO, if all you need is "routine" service work, like periodic oil changes, or a wheel alignment, ie. work that you must pay for anyway, take it anywhere you like, including an independent garage or even a quicky oil change place. If the latter, you should probably bring your own oil (Mobil 1).
The advantage of a "drive thru" type of service place is that you can usually be present while the work is performed and thus keep an eye on your car. If the work required is something more serious or an actual repair, take it to a "certified" Corvette Chevy dealership. Call CAC to find one in your area. This means they have a properly GM trained repair technician in the service dept.
Call the dealership to make sure they have such a tech and that he is certified to work on the C5.
As an alternative, check with a local Vette club or other owners in your area and find out which dealer(s) they recommend.
Call and make an appointment. Don't come across as a difficult to please Corvette customer with some kind of an "attitude". Just be polite and professional about having the work performed by a
qualified mechanic. Try to get to know him if at all possible and at the very least find out his name.
Simply explain to the service writer that you want to be kept informed of the progress on the work on your car, such as to the outcome of troubleshooting and parts necessary to fix the problem. Politely mention that in return for being kept informed and ultimately being satisfied that your problem is taken care of, you'll be glad to give the service writer a high rating on Chevy's customer satisfaction survey.
I think most "authorized" Chevrolet service departments, that are actually reputable, are sufficiently motivated to get good marks on that customer satisfaction feedback survey, so use this as leverage to get the best possible service.
