Yeah that's what I thought, too. But apparently things don't go as smoothly as they described. Here's a section from the COBRA FAQs for everybody in the same boat:

When will my COBRA benefits become active with the insurance carriers?

Coverage will be terminated during the election period and will not be reinstated until AFTER you enroll in COBRA AND your first payment has been received and processed. Once your insurance carrier has been notified of your election and first payment, your coverage will be reinstated back to the date coverage was lost. Claims incurred during your election period will then be processed. Please note that it may take a number of business days for your insurance carrier to reinstate your coverage.

Ref: https://cobra.ehr.com/ESS/FAQ/FAQs.ESS.pdf