If you accidentally visit their page with the wrong browser (Safari or Chrome, I forget) when you need to renew an expiring client certificate - the browser doesn't download it properly, you can't ever request another one. Anyway, letsencrypt sorts that out.
I don't know what you have been doing the last 20 years on the web (and I'll assume it's more than just surfing facebook), but it's not entirely uncommon, and I've encountered it several places.
Symantec's CA uses it. My online bank used to do so too. I've seen VPNs using it. Iirc some IPv6 tunnel-providers also require you to authenticate using certificates before letting you set up new IPv6 subnets.
It may not be mainstream, but it's part of the standard. And it's much more secure than a regular username/password, for the same reason SSH keys are more secure than allowing username/password logins.
To be clear about that: My point about half-assed was your seeming unwillingness to back up client-certificates which gives full access to your real certificates and (in some cases) private certificate keys.
Unless on Windows (where StartSSL has its private keys marked non-exportable in the certificate store, sic), doing such a backup takes almost no effort. There's no excuse for going all the way through to get a cert and then not bothering backing up these client-certs too.
./letsencrypt-auto renew(Browser makers are quarter-assing their UX for using client certs, but that's a separate issue).
So you want a secure website, and you agree that SSL is needed for things to be secure.
But you're not willing to put in one inch of effort yourself to secure your own SSL keys. You can't even bother to back up the master key to your own certs, because it's too much work?
Cognitive dissonance much?
If you care about security, then do it properly. If you're going to do it half-assed, just don't bother at all. All you're doing then is contributing to security-theater, which is all the work and no real benefits.