Are you pairing computers by copying certificates to visit this site?

Touche.. actually a good point, but actually those are two different situations. With one, I'm accessing a website and trusting that the certificate is signed by someone I trust; so the trust in my browser certificates (which include certificates from hundreds of certificate authorities all over the world, any one of which could be compromised, robbed, or controlled by an adversarial person or even government) is extended to the site that I'm visiting. To say this is weak sauce rather understates how bad this actually is. (To paraphrase Churchill, this is the worst possible design, except for all the rest.)

With the other, I'm logging into a server for the first time (and I could simply deploy the same trusted host key to all my ssh servers via an autoscaling configuration or whatever). I think it's debatable if TOFU is worse or better than your (granted clever) metaphor.

(to those who'd recommend userify, yes - great for the client login issue and definitely increases security, but to parent's point, TOFU is still needed unless you want to distribute host pubkeys)

Pairing is absolutely necessary for bidirectional authentication, where each party must verify the identity of the other end.

To visit this site, there is no pairing, because the site does not know who I am.

In order to verify the identity of the HN site, I must trust that the maintainers of the installation packages of the browsers that I use (Firefox, Vivaldi, Chromium) have ensured that the built-in certificates have reached me through a secure path. This actually requires much more trust than when someone answers "yes" to the SSH unknown host message.

If I use certificates for accessing e.g. the network of my employer, then my work computer must be paired with some corporate server, i.e. a unique certificate has been generated for myself and it has been copied to some certificate authority server for signing and then to my computer, and also a certificate of the local certificate authority has been copied to my personal computer.

While pairing is unavoidable for bidirectional authentication, it is not necessarily direct between the end points. Both end points must have been paired with at least one other computer but they need not have been paired between themselves previously if there exists some path through secure connections that have been originally created by pairing.

When certificates are used, usually the pairings are not done directly between end points, but each computer must be paired with the server hosting the certificate authority.

The term "pairing" is not used frequently, but it should have been preferred, because frequently the users do not understand which are the exact actions on which the security of their communications depend, which leads to various exploits. The critical security actions are those that perform the pairing.

"Pairing" of 2 systems, e.g. A and B, means that some information must be transmitted through a secure channel from A to B and some other information must be transmitted through a secure channel from B to A. An alternative pairing method is to generate both pieces of information on one of the 2 systems and transmit both of them through a secure channel to the other. The information exchange channels must already be secure, because before pairing authentication is impossible.

The pairing between a PC and the server hosting the certificate authority can be done in various ways, depending on where the PC certificate is generated. If the certificate is generated at the certificate authority than both it and the root certificate must be copied through a secure channel to the PC. If the certificate is generated on the PC, it must be sent through a secure channel to the CA for signing, then it must be sent back also through a secure channel.

In practice, administrators are not always careful enough for the channels through which certificates are copied to be really secure. For instance they may be copied through network links that are not yet authenticated, which is equivalent with the TOFU method optionally used by SSH.