This is a very common practice because there is no contract between the health plan and the provider, the plan then reverts to applying contract provisions between the plan and the member and most plans indicate the benefits belong to the policy holder and therefore they are issued the payment.
Some of this issue was resolved with the CAA of 2021 Section 102 "No Surprises Act" but the provisions are limited to emergency services or when a patient received care from an out-of-network provider at an in-network facility without informed consent or prior authorization (a physician's office doesn't count as a facility) then the benefits are to be processed at the in-network benefit level and payment is supposed to be issued to the provider. Rarely does this happen with an independent lab. However, if you are billing for a pathologist who provided emergency services or provided the services to the patient at an in-network facility the No Surprise Act.
The patient is not obligated to forward the check from the insurance company to an out-of-network provider, they should use the funds to pay their bill with the provider, but they are not legally required to. However, there is nothing that can be done to force the patient to pay their bill to the provider, just like you can't force anyone to pay many of their bills. You can try to collect from the patient and if they don't pay then if your state allows it, you can send the patient to a collections agency to attempt to collect on the bill.
I know this is frustrating for out-of-network providers who go unpaid because the patient does not pay their bill, but the patients are frustrated that their provider utilized and out-of-network provider for services without their knowledge and the patient was not offered the opportunity to choose an in-network provider. No one wins in this situation.