Lets assume the original was VHS tape, the number of pixels on VHS tape is 330 pixels wide which would be a pretty small area on any computer screen these days. I would expect the box has done its job and you've got as much information off the tape as you are going to get.

You got it right comparing to youtube as the orignal resolution of youtube was about the same as VHS -- although it is increasing as they get access to more bandwidth.

The trick is to get the viewer to upscale the image when you view it. You need to put the viewing app into full screen mode -- be prepared for a pretty fuzzy picture -- your still getting all the information that is on the tape but it is spread out a bit more.

Each viewer -- Window Media Player, Quicktime, Real, etc has a different way of switching to full screen mode.

Using some video processing software to upscale the video and save it might give a better result that live upscaling, probably not by much though.