FALSE.
You have to shell out the money for the test, and then apply for reimbursement through the private company. You may or not get the money back, because that is at the whim of the private insurer. Many private insurance companies are going to get their executives rich on this nonsense.
I assume it will work the same as getting prescriptions. You go to the counter get the home test kit and then their is a determination if your private insurer covers the item (or you need to provide a co-pay, etc.). If your insurance company does not cover the home test kit when the pharmacist looks it up then most people will just leave it and walk away.
I don't think that anyone will be willing to pay for the test and then try to submit for reimbursement from their private insurance company. It's not like they do this for any of the other drugs or items covered by insurance (beyond their co-pay which is paid at the time you pick the item up).
I will note this requirement that private insurance companies cover home test kits is not reimbursed by the government -- the cost will be borne by the insurance companies.
U.S. to not reimburse private health insurers for covering at-home COVID test costs
https://www.reuters.com/world/us/us...covering-at-home-covid-test-costs-2021-12-02/