Thank you for the advice. We followed up with our OTA (MakeMyTrip.com) but they were unable to rebook us on other flights that showed with available seats on emirates.com - they said their system showed those flights with the status "Waitlist closed." However, Emirates has now updated our itinerary to put us on a flight FZ446 from BOM-DXB on Jan 5 - so we will make our DXB-SFO connection, albeit with a 24 hour stopover in Dubai. There are still a couple of strange things:

1. FZ446 shows the status "Flight does not operate" when I look at our itinerary on the Emirates site, even though it looks like it's a daily flight.
2. The original flight on which they had rebooked us (leaving at 10 am on Jan 6) still shows in our itinerary, so we now have two BOM-DXB flights in our itinerary

I have no idea why they didn't simply rebook us on another EK flight on Jan 5 rather than FZ. These new seats are also in economy, which is also strange.

I'm hoping it all gets sorted out eventually, but at least we now have a viable flight showing up.