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.
I've always wondered if it's better to write at a blog (overcomingbias.com) or write under your own name (scottaaronson.blog).
The benefits of writing under your own name seem to be easier credit to your work and that people get an instant snapshot of what you're interested in. It also shows others how much you write and how long you've been writing for, which is a display for consistency for potential employers/partners. The cons of this, though, is that sometimes one feels they can't share personal projects they're working on or other "less organizable" stuff.
If you write under a blog name, it seems to feel more credible (not sure why this is). It also allows you to use your personal website to talk about projects you're working on things you're researching, while having the option to link back to your blog to showcase your writing. If you do this, though, the consistency factor isn't there like it would be if it were on your personal website.
I currently write at daltonmabery.com, but have debated changing it to 221b.blog (a nod to Sherlock Holmes), colloquim.blog, or something of the like.