Two health insurance policies question

Hi everyone, my wife and I both have health insurance through work through two different companies, and with the birth of our child, we decided to go onto her insurance. My work gives me a cashout if I don’t use all the money provided for insurance coverage in an HSA, but only if I selected a plan with them. My wife’s deductible/out-of-pocket max is much better, along with not having to go for referrals. So I selected the cheapest plan for my work and got the family plan with hers.

Essentially we are trying only to use my wife’s insurance to consolidate our charges into one out-of-pocket max but still keep my insurance for the benefits from work. I received a Coordination of Benefits paper today from my wife’s insurance. After reading about the process, it sounds like they would send the bill to my single coverage insurance first and then apply it to the family plan. If I only presented our family plan at the doctor’s office, would those bills still be sent to my single coverage first before the family plan?