About six months ago I went to a government auction site that required Internet Explorer. Yes, Internet Explorer. The site was active, too; the auction data was up-to-date. I added a user-agent extension in Chrome, switched to IE, retried and it worked; all functionality on the site was fine. So yeah, I was both sad and annoyed. My guess is this government office paid for a website 25 years ago and it hasn't been updated since.
Not really. You can access any Korean bank or government website using Chrome, and they actually recommend Chrome these days.
They still want to install a bunch of programs on your computer, though. It's more or less the same stuff that used to be written as ActiveX extensions, but rewritten using modern browser APIs. :(
Yeah it's probably an ancient web site. This was commonplace back in the day when Internet Explorer had 90%+ market share. Lazy web devs couldn't be bothered to support other browsers (or didn't know how) so just added a message demanding you use IE as opposed to fixing the problems with the site.
In South Korea, ActiveX is still required for many things like banking and government stuff. So they're stuck with both IE and the gaping security hole in it that is ActiveX.