It's not as if Caltrain is required to run at grade. It's just a historical artifact of how the system evolved. If we had the money and political will to replace Caltrain with BART, we'd have the money and will to create a grade-separated Caltrain.
What would be the advantage of BART over electrified, grade-separated Caltrain? I'm hard-pressed to think of one, and there's definitely one major disadvantage: the nonstandard rail gauge.