You got that prompt because the server certificate provided by the website and given to your browser has a PKI trust chain consisting of an intermediate certificate authority and a root certificate authority, both of which are owned and operated by the same organization running the website which provided it. I can see the humor in it.

It's like asking somebody to pinkie swear to make sure they are telling the truth.

All you trust by default is whatever the browser creator chose to include and whatever Microsoft wants you to trust. In that sense, the .mil page with the big red privacy warning is more private than anything else you look at online. I look at it like the page is saying, "I don't trust anyone else to tell you if you can trust me".

It makes no sense from the outside because you have the ability to chose to trust it(The government and therefore it's network resources) or not. but for gov/mil personnel and Sysadmins, it's indispensable. It forms the foundation for client/server authentications of all kinds as well as network traffic encryption and email encryption and digital signatures and code signing and provides a mechanism for two-factor authentication of individuals. There's nothing even remotely close to DoD PKI in the world and I would have no problem adopting it's design for general use if it was available and readily used by others.

Although I am a little bias...This is my job.

There is a program called "InstallRoot" provided by DISA which will install the root certs(2. The current and the new SHA256), intermediate certs(17 but dropping to 15 this Wednesday) as well as the test chain and remove 2 certs which bridge military networks with fed-gov networks(no thanks). It will do this for Windows Certificate Store as well as Firefox's bullshit cert mess(Good intentions, bad execution). It obviously only does this for unclass networks.

Trust certificates are widely misunderstood. Having malicious trust will only allow you to make a mistake. You can have asymmetric key algorithms using TLS_ECDHE_RSA_WITH_AES_128_GCM_SHA256 in perfect forward secrecy and still chose to trust someone you shouldn't. It only allows you to be more confidant in knowing who they are.