Turns out there is lots of sequence-based tools I didn't know about - see here. From what I can tell by briefly inspecting their main pages, all of them are based on position-specific features extracted from PSI-BLAST or similar searches. That also doesn't scale well with tens of thousands of proteins.
If I were to start this project, I'd try to develop a classifier based on some approach that can generate position-specific features quickly, as that appears to be the bottleneck. Something like this or this could work.