I don't know that I'd call it "dumbed down" necessarily, but the minimum power/resources required for capturing video in standard definition was surpassed years ago now. Anything that is 2Ghz or faster, P4 or AMD, tends to work perfectly fine. Edge goes to Intel, however, when using certain tools that require SSE2 or SSE3 CPU instructions -- something missing from many AMD chips.
The biggest reason to get an older system is for the coveted AGP slots, using one of the highly-regarded
ATI All-In-Wonder Radeon AGP cards. Guides, of course, are at:
http://www.digitalfaq.com/guides/video/capture-ati-mpeg.htm
http://www.digitalfaq.com/guides/video/capture-ati-avi.htm
The second reason is that you can build a box very low-cost, and without "messing up" your normal computer. Or inversely, you can create a system that is free of the mess often found on a person's regular computer (and that includes the Internet!!).
The Precision 380 appears to be a dual-core system with PCI Express bus -- so that probably isn't the best option for a capture box, if you were looking towards those ATI cards. "Isn't the best" as in "won't work".
The only thing to maybe avoid is the VIA line of motherboards, that were often used on cheaply-made AMD and Intel Celeron systems. Aside from that, stick to normal good advice. Meaning a good board is one from Intel, Asus, nVidia, ... and that other brand I always forget (the one nobody's ever heard of).
There is really nothing for or against the slot types, AGP vs PCI capture cards, it's all about the card itself. A Matrox PCI card is going to be better than an ATI AIW AGP, but a Hauppauge PCI is not as good as the ATI. It's not really about the slot.
You are wise to consider the whole enchilada here. It is "DIGITAL VIDEO" after all, don't forget the "digital" part! The computer and capture card can be just as important as that VCR.