Both types of names are forms of branding. Your real name is your career's brand, while a pseudonym brands an idea. Pseudonyms attract more attention because ideas are more interesting than somebody else's career. These purposes don't exclude each other, but they signal different priorities.

Overcomingbias.com is an interesting example, as I'm not sure which one it is.  It started as Robin Hanson and a few friends, shifted to Robin and Elizer, and now is pretty much just Robin.  It's never been open to newcomers, always a private space.

I think that's important - there aren't just two choices.  There's a continuum (or really, a multidimensional space) of options with different levels of control, exclusivity, visibility, effort, etc.  You don't even mention traditional publishing - getting pieces into periodicals, or running one yourself.  

Which brings up the next topic - these are not exclusive options.  Certainly not over time (you'll often use one medium to start, then add or replace with others as you and your audience(s) learn what works), but also not at a point in time - MANY LessWrong posters run their own blogs and link or crosspost to LW.  Many magazine or book authors also have a twitter stream and/or substack and/or their own space.

So: Both.  Neither.  Try other things too.  If you like writing, write.  If you want to make money at it, still write, but experiment with different monetization schemes.  Expect things to change over time.

Practically, this last element is important.  Everyone I know who's been doing this for more than a few years keeps private archives to prevent loss/rot of their work as platforms come and go.  Those that didn't start out with such a mechanism regrets the loss of some of their early writing.