Who should verify health insurance coverage and if procedures are approved?

In Illinois (and the entire US, possibly), who is responsible for verifying health insurance coverage (if doctor is in-network) and if any procedures done (echo, stress test, blood draws, etc) are covered? Should it be the patient or the doctor's office? Shouldn't dealing with the professionals in the insurance company be done by professionals in the doctor's office, instead of the patient who sometimes does not understand half of how the health system works?
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It's your health, and your bank account that will be affected, so you should be the one checking the coverage and procedure approvals. You don't have to understand how the entire health system works, you just should be capable of asking an insurance care representative, or your HR representative, whether procedures are covered, and how much.
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