While I agree that it seems pretty strange happening on three different machines like that, it appears to be a somewhat common problem based on other forum posts, product reviews, and such. So, maybe it's not that strange after all.
A few updates:
1) After my last post to you all, I did a little more work on the DirectX side of things. I had actually found the exact same link you posted and tried installing from there. The install went smoothly, but nothing changed when running the capture app. I even tried a complete uninstall of the Roxio software with Revo, then a complete reinstall after the DirectX installation, but no luck.
2) My machine definitely meets all of the aforementioned requirements. Also, when I said that I tried to run in various compatibility modes, that was [a] only after all other approaches failed and only when running the capture app. Normally, I shy away from compatibility modes as much as possible. Win 7 had just suggested it after one of the app launches failed, so I figured it was worth a try that one time. I've since gone back to just running normally. My user is an admin on the machine(s) in question, and I've also tried "Run as Administrator...". Same result either way.
3) Strangely enough, I also noticed that on at least one of my machines, a number of things stopped working after the Roxio installation. Specifically, Spotify hung when I tried to play any music. Same thing with a few games. I also tried multiple restarts after each failure, just to make sure the installation process and/or a failed launch of the app wasn't gumming anything up. Of course, when I uninstalled the Roxio stuff and restarted, Spotify started working again, as did the games that were hanging. Shrug.
Unfortunately, its starting to look like this has basically been a complete waste of time and money. Even when I was able to get the software running (on that XP machine I borrowed), I BSoD'ed every time I captured a video. So, this VHS-capturing adventure has pretty much been an epic fail.
Thanks so much for all of your help thus far, everyone. If I find anything out, I'll be sure to update this post for anyone else facing the same issues. I'm running out of time before the holidays to get this project done, so if nothing else, I might pursue another piece of software with the same device. Sucks that I had to pay for the Roxio software if it's unusable for me, but at least I might be able to salvage something out of all this before the holidays.