Basically all browsers default to follow the operating systems appearance settings. I don't know why you're specifically asking to not change the OS settings, why would you prefer dark mode of your OS when you want the websites to be light mode?
To answer your question, I have no clue about Safari, but in Firefox you go to the settings page and right on the start page there is Language and Appearance where you can select your theme.
> when viewing specific websites that are hardwired to use it,
Yes, that is an issue. But that is not an issue of dark mode per-se, it is an issue with software quality and design decisions themselves. Some apps might implement their own theme switcher (which they should not do, but people seem to like making their own worse implementations of browser standards), others might not implement a dark or light mode to begin with.