I created a service using Automator called "Open URL in Public Window". Its purpose is to never open a selected link in a private window (which will happen if the current front-most Safari window [before opening the new URL] is a private window).
I'd actually prefer all clicked links(/highlighted URLs) to open in public windows, and have that be the default behavior of the "Open URL" service that shows up when you right-click a selected URL or link, but I cannot find where that service exists so that I can edit it. Maybe it's not even editable or even a Quick Action/Service.
I'm not even certain that the "Open URL" service I see in a context menu from right-clicking a link in other apps (e.g. Mail.app) and going down to the "Services" sub-menu is a standard service (did I create that a long time ago and forgot about it? - regardless, I cannot find it in e.g. ~/Library/Services).