Patients absolutely carry the right the determine if they or the insurance company are the payer of a particular service. I'll concur with other posters that patients reserve this right, just as the insurance company reserves the right to call a particular claim "non-covered" and make that patient responsibility.
As for the patient withholding information from the insurance company, that is a matter of contract that exists between the patient and the payer--and likely does not involve the provider. I am not saying they don't exist, but I have never seen a provision in a provider-payer contract that stipulates all services must be billed to the insurance carrier (minus Medicaid, which is mentioned in full detail above).
I'm sorry, but this is a non-issue. If I, as a patient, determine that I want to pay out of pocket for the services from a particular provider or all my care, that is my choice and financial decision. Here's another great example: I once encountered a patient who wanted private pay for all services within the 4th quarter of the year. When someone finally brought this up to the patient, in a nice, neutral way, the patient indicated she did this to positively impact her taxes--spending greater than 7% of your annual income on health care costs apparently is or was a substantial deduction.
It's not much of your business, so long as you respect the patient's wishes.
Kevin B. Shields, RHIT, CPCO, CCS, CPC, COC, CCS-P, CPC-P, CPC-I