I'm kinda surprised you've managed to be on HN for 5 years and never come across the concept of a "LAN" or "VPN" before, but I guess you're one of today's lucky 10000. To the first, sometimes you have machines (or VMs) local to your own network but in another physical location that you'd like to be able to access from your own system. It's a fairly significant use case, and one where no internet connection is involved whatsoever. For example it's generally desirable to locate powerful (and in turn generally loud) servers and associated gear (including environmental control, redundant power etc) in physically isolated locations from where the humans are working for noise reasons if nothing else, though security and efficiency are important as well. While it's possible to pipe raw video over IP, a quality remote desktop solution will generally be more flexible/scalable and doesn't require special (expensive) extra hardware and potentially additional fiber.

And for systems located on other LANs remote from your own, you can use a VPN to link them securely as if they had a direct physical (though higher latency/more jittery) link, again avoiding any exposure to the public net. That then reduces to the above. In both cases it's desirable to have zero unnecessary 3rd party dependencies.