> I think the LE endgame is that setting up HTTPS will be as easy as setting up SSH
It already is. Use a self-signed cert and ask your visitors via a side channel to tick the box to add an exception. That's the security model of SSH, and has the same assurances. It's just a terrible comparison. We can do better.
A better comparison would be with DNSSEC...
The way to simplify TLS deployment, while getting a better level security as DV CAs, would have been DANE. DNSSEC has an unfortunate reputation as being hard to setup ("pdnssec secure-zone mydomain" is not hard), but it also has exactly the same X day rollover key problem to solve as LE. Solving it once instead of twice would have made a lot of sense.
The symmetries between DNSSEC and LetsEncrypt are striking. Your account key is your KSK. Your cert is your ZSK... but instead of your webserver (in DNSSEC, your DNS server) doing rollover for you, you're depending on a pile of scripts, leaning on an already insecure DNS, and setting it up is still, comparatively, a pita.