I've tried both methods here on Windows 8.1 Update 2. At first the symptom had been the same as the OP, except that I was doing exactly what the OP did as his solution -- extracting the files and running Vs6sp6.exe.
So I tried the other solution, installing MSDN. After MSDN installed, I tried SP6 again. This time, it tells me it can't install because it can't find any VS6 products installed. But VB6 (without SP) still starts up fine. Also MSDN starts up normally.
Any suggestions? If I have to run it in a VM I will; I already have Windows XP on a Hyper-V VM for the original early 90's Oxford English Dictionary. But it would be a lot more convenient if I could get it working directly in 8.1.
BTW I've been running it fine in Windows 7 for years. There are some hoops to go through to get it installed, and it changes the screen when it starts and stops, but overall not bad at all.