Update #3: It's fixed now!
I called Schwab support at 800-435-4000. The first rep had me try everything in Chrome—still no luck. Then we cleared Safari's history and cache, but that didn't help either.
He ended up transferring me to a Technical Support Specialist. The specialist went through a bunch of the same troubleshooting steps, then finally said something like, "Hold on a sec, let me reset and re-sync your account on our end."
Safari suddenly started working perfectly.
So yeah—turns out it was completely on Schwab's side, not a browser issue with Safari or Chrome at all.
A side note, Firefox always worked with Schwab.com. Firefox was just a hassle to use, I prefer Safari.