Get the papers out. Don’t focus on an arcane document that will gather dust for the next 50 years until the departmental office needs space and throws the old ones out.
I didn’t even get the bound copies of my thesis. It’s a PDF sitting on a department server (it’s open access!).
Thus, this image of a thesis as a big block of paper, gathering dust in a hard-to-reach library section is out of date. It is a document, widely available, sometimes even read. During my graduate studies, I did read a few theses by others.
Because things are now electronic, people will read your thesis. Probably not PIs (who will read the executive summary, ie, the papers), but graduate students will (and, incredible though it may seem, graduate students are people too). The median paper is probably read by very few people too, by the way.
Have your cake and eat it too.
I did, however, enjoy being able to write without any page limits and put in more details than made it into the final version of the papers. I also put in some side comments that I thought were cool but on the periphery of what the chapter was about. In some ways, I had the opportunity to write down the type of thought I would now cut out of the paper and put on the blog.
This is not to say that you shouldn’t publish in peer-reviewed outlets. You should (as I said, my thesis is a staple thesis, almost all of the content is available in peer-reviewed literature too). But we shouldn’t use bad or out-of-date arguments, like nobody reads your thesis, for it.
You should publish because your résumé looks better with publications than with a thesis. Theses are like assholes, everybody has one (everyone who is applying to the sort of academic jobs we are discussing).
So, write your thesis and publish.