I've heard of that rule before (explained to me in the exact same way you explained by one of my previous doctors), and it definitely worked in a similar way for me in the past too. But for the past few years, my process has been much easier and simpler.
My insurance provider (Premera) has their own mail order pharmacy (called Express Scripts[0]; technically independent, but Premera officially partnered with them), so i dont have to go at all. My doctor writes a prescription, sends it directly to the insurance, opting for the mail pharmacy option checkbox, and I get my meds in the mailbox. They arrive all at once, with a single pill bottle containing 90 days worth of medication.
Just in case it varies by state, that's how it works for me in WA.
0. https://www.premera.com/visitor/prescription-savings (the link talks about Express Scripts specifically on that page, it isn't just a generic Premera page)