Why I need a browser that's willing to accept bad TLS certificates
One of my peculiarities is that I absolutely need a browser that's willing to accept 'bad' TLS certificates, probably for all species of bad that you can imagine: mismatched host names, expired certificates, self-signed or signed by an unknown certificate authority, or some combination of these. There are not so much two reasons for this as two levels of the explanation.
The direct reason is easy to state: lights out management processors. Any decent one supports HTTPS (and you really want to use it), but we absolutely cannot give them real TLS certificates because they all live on internal domain names and we're not going to change that. Even if we could get proper TLS certificates for them somehow, the cost is prohibitive since we have a fair number of LOMs.
(Our ability to get free certificates has gone away for complicated reasons.)
But in theory there's a workaround for that. We could create our own certificate authority, add it as a trust root, and then issue our own properly signed LOM certificates (all our LOMs accept us giving them new certificates). This would reduce the problem to doing an initial certificate load in some hacked up environment that accepted the LOMs out-of-box bad certificate (or using another interface for it, if and where one exists).
The problem with this is that as far as I know, certificate authorities are too powerful. Our new LOM certificate authority should only be trusted for hosts in a very specific internal domain, but I don't believe there's any way to tell browsers to actually enforce that and refuse to accept TLS certificates it signs for any other domain. That makes it a loaded gun that we would have to guard exceedingly carefully, since it could be used to MITM any of our browsers for any or almost any HTTPS site we visit, even ones that have nothing to do with our LOMs. And I'm not willing to take that sort of a risk or try to run an internal CA that securely (partly because it would be a huge pain in practice).
So that's the indirect reason: certificate authorities are too powerful, so powerful that we can't safely use one for a limited purpose in a browser.
(I admit that we might not go to the bother of making our own CA and certificates even if we could, but at least it would be a realistic possibility and people could frown at us for not doing so.)