I feel a bit dumb. Phil Scott and Nic Wise both pointed out in comments that my complaints about Direct3D not working well under Windows 2003 were easily addressed by running dxdiag and enabling DirectX acceleration. I had remembered the other necessary part (which turns on acceleration at the OS level) but not the part where you need to turn it on at the DirectX level. (The FAQ they pointed me to contains some other useful details, too.)
What’s funny is that I must have done this the last time I installed Windows 2003 – I just forgot half of the process. Oh well, perhaps my ignorance will help someone else figure out how to play Quake on their work machine. And the good news is I still have good reasons to have bought my new computer.