Sorry, but... why not dual boot XP on your 1GHz PIII system? Assuming you have enough RAM, that should run quite nicely if you want to run early 2000s games...
I guess I am just struggling with your 2005-6 timeframe. That was the darkest era in Intel's history, a time when Intel fanboys like me almost got AMD X2 3800+s but were only stopped from going to the dark side by the rumour that a magical new processor would be coming from Intel that would make the world right again... (and of course, that processor shipped as Conroe in summer 2006, AMD was knocked out of the running for a decade, and all was well in the world). If you wanted a period-correct enthusiast gaming system from 2005, you want a 3800+ on 939.
If you're willing to go into early 2007, then there are going to be plenty of options in the 65nm C2D/C2Q category. But my understanding is that most businessy desktops remained on HotBurst throughout late 2006, at least the more value-priced ones. Intel had big production constraints on Conroe and one of the ways they dealt with that was continuing to sell cheap hotbursts to large OEMs.
Other observations I would make:
1) This happens to coincide with the time that good GPUs really started requiring additional power, dual slots, etc. Sure, there were still some single-slot cards in 2006 (I think my 7900GT was single-slot, though it needed PCI-E power and was looooong), but the days of video cards that could fit into anything were ending around that time period. And if you're wanting to go with a desktop case, that means a low-profile video card, and... ouch, the options there just get worse and get worse. Also, not sure how many sound cards are low-profile friendly; certainly I've never seen a low-profile Creative Labs card, not to say that there hasn't been one.
2) I don't see a reason to go AGP - a DDR2/SATA/PCI-E system will save you lots of money. People wanting to run 98SE have been driving up the price of all things AGP.
3) I think finding OptiPlexes/ThinkCenters/I-forget-HP's-brand-name from this era in 2024 will be trickier than finding random white box or home-grade systems. Most of these systems would have been bought/leased by corporate buyers who would have replaced them around 2009 or so, then they would have been sold as 'refurbished' back then, who knows how many good ones are still available now. Now, I do think there will be a huge number of late-2000s C2D/C2Q systems still around finally getting tossed due to the Windows 11 requirements, so... that might create some bargains.
I actually had a system of the sort you're thinking about for a little while. It was a ThinkCenter something or other in an SFF desktop case; some kind of Pentium D 9xx. I got it thrown out by an employer without a hard drive for $0, put a random SATA hard drive in it (had to buy the Lenovo mounting hardware first), maybe put some random RAM I had floating around, and then ended up selling it to a Mac-using friend who needed a Windows machine for some work thing. Completely unmemorable hotburst system, I would say.
The final point I would make - you're not buying Voodoo 3s or other legendary vintage components for top dollar here, so... you don't need to do a ton of research and worry about getting it wrong. I would guess your mid-2006 HotBurst business desktop should go for like <$50.
