lol $350 to build a machine? That will buy you the CPU and motherboard, maybe. Though prices in the states are quite cheap, so maybe it could be done... don't forget to account for shipping and taxes though.
If you are going for a high performance machine like say, i7 with 16GB of ram and a SSD, it will be cheaper to build than to buy because OEMs will price higher end stuff at a premium, but if you're going for like core i3 with 4GB of ram and slow HDD type machine then it will be cheaper to buy. Personally given your requirements of future proofing I'd go the higher end route, but you're looking at close to a grand per PC.
The challenge is trying to find 30 of the same parts though. I think it would be cool to see more organizations custom build but sadly it seems the industry just makes it hard to build systems in bulk mostly because of part availability. Most sites even have a limit per person when buying certain parts, or have low stock and next stock will be another revision of the part. This makes OS deployment harder as you need to make multiple images instead of just 1 install then clone to all the PCs.
I really like the custom build approach though, but you definitely need to up your budget, and part sourcing WILL be a challenge.