will do on the cable.  It should be here early this week.  We'll see if it's a $10 solution or not.

Sorry I took so long to get to this, I'd been meaning to respond, since this is actually something I actively do and it works well.
 
For capture hardware, what you want is something from Yuan High-Tech or Startech. Startech has their stuff made by Yuan High-Tech, so it is the same hardware, same drivers. They have excellent drivers with all the adjustments you might need and they export the device as a standard directshow device so everything can see it and use it no problem.
 
For software you can use anything that knows how to talk to a standard video device (like a webcam) but I recommend using Open Broadcaster Software. It's free, and does the trick nicely. You set up a project in 640x480 which is the square pixel version of the NTSC standard (you could technically go lower using VHS) and have the video from the capture unit fill the whole screen. Then just have it archive to a file (turn off streaming). Setup your encoding settings to preference and based on what you have in your computer.
 
Using that records tapes like a charm. OBS also knows how to use Intel QuickSync and nVidia NVENC so you don't need a huge processor for encoding, though in general for things that low rez x264 cranked up gives better quality per bit and is unproblematic to run.

Sycraft, what capture hardware would you recommend from those companies?

The specific ones I have experience with are the Startech USB3HDCAP, Startech PEXHDCAP, Yuan SC542N1, and Yuan SC510N4 SDI. All of them have the same driver interface and work with all the video capture tools we've tried (Skype, OBS, and Panopto primarily).

I did that years ago by stepping through a video camera. Source > Camera > Vegas from the camera firewire port. I also used a video time base corrector for the problematic ones. Lots of time and hassle but it worked. I'm not sure I would go through all that again though.

Well....my little $8 video cable (Composite/Svideo > USB) works.  I have to use the software that came with it, and I can play back the VHS player and it will record in real time to my hard drive.  Sony will not recognize my USB adapter.  It shows on the available video input list, but cannot open it.  It's an "EasyCAPture" device...purchased of Ebay.....definitely Chinese, as the directions are poorly translated to English as well.  Will see how it behaves on a 1/2 hr video next.

Just created a 4 gig video/audio file off a 1 hour VHS tape, using this little cable.  Worked fine and the quality is no worse than the original tape.