That does sound like an odd issue that someone will provide a good suggestion on.
I know this is not what you asked for help with, but here is a suggestion that will open Safari with "A new private browsing window". A Private browsing window will clear all cookies when you exit the browser and therefore you will no longer need to clear cookies manually:
- Click Safari menu
- Choose "Preferences"
- On the "General" tab
- Set the "Safari opens with" setting to: "A new private browsing window"
Note that private browsing features of all modern browsers help to keep your local browser clean of most browsing history (and cookies). However, the unfortunate fact is that modern browser fingerprinting techniques have made it nearly impossible for anyone to prevent websites from tracking you. Clearing cookies, ad blockers, and privacy extensions do nothing to prevent tracking using current techniques. Until browsers start building in far better tools to help users protect privacy, we are all being tracked no matter how privacy savvy we believe we are.