I was enrolled at the distance university of Hagen in Germany for a while and they require the use of client certificates for access to their online portal. There were clear instructions for how to create and use a client certificate but I suspect they have an advantage in that many of their students enrol in technical subjects or already have job experience. Compared to an ordinary website they also have the advantage that students HAVE to use the website and they're the only public distance university in Germany so there's no competition.
From a user perspective the client certificate is incredibly cumbersome. It's a file on your computer, so you have to remember where you put it and move it to new devices if you want to use it there too. It also means you're more likely to misplace or lose it though you're probably less likely to leak it compared to a password.
The instructions also largely boiled down to "use Firefox". In Germany Firefox has a huge market share and is widely deployed as the alternative browser in the public sector (although IE still exists due to contracts with intranet service providers). In other countries things look differently.
In Chrome the experience of using client certificates was even more convoluted and the university officially didn't support Chrome because apparently client certificates flat out didn't work in Chrome until fairly recently (i.e. a few years ago).
In terms of UX, creating and using password is trivial compared to creating and using client certificates. Of course this is mostly because most people do passwords wrong. Creating and using a secure non-guessable password is difficult (though services like 1password or lastpass have made it easier at the cost of adding a single point of failure) but it's still marginally easier than creating and using a client certificate.
The big difference though, is that insecure-by-default is not as big of a cost to a website or software as the bad UX of client certificates. Sadly the UX of client certificates likely won't get better in browsers unless more sites use client certificates -- so it's stuck in a Catch 22.