Is the site broken?
Unlikely, it is more likely that the consulate in San Francisco is not booking appointments at the moment. Attempting to book an appointment using the link you gave, gives a screen like this...
And on this page we observe that French consulates use an app called 'rendez-vous'. So switching the location to Vancouver gives us this page...
which shows that the app is working. To confirm, let's see if the same user experience occurs in Rio...
Rio works. But there's no indication that San Francisco is broken. So we can put this down to one of several alternatives...
The San Francisco consulate is not staffed up at this time in a way
that can accommodate your appointment request (i.e., the app is reporting the situation accurately);
A consular staff member has corrupted the database such that all the
Vice Consuls in SF are shown as indefinitely busy;
The San Francisco consulate has somehow fallen off the grid and your
experience is the app's way of telling you.
To move forward, you can pursue one of several options...
Contact the consulate using their 'contact us' details and explain
what the problem is. It's likely this may result in a problem ticket being raised at the IT support facility in La Defence (that's
a bad thing).
Book your appointment in Vancouver (or another site with available
slots) and then put in an immediate change request on it (or
alternatively, just attend the appointment).
Play 'map-and-die' with the URI tail to locate a different consulate
more convenient to you (that's how I found Vancouver and Rio by the way).
My personal favourite: rework your itinerary such that you will land in a different member (thereby changing the cognizant issuer).
Keep trying and trying to wait for the San Francisco site to have
Start (very tactfully) haranguing them on Twitter about it. You can start by asking them to identify your liaison for 'rendez-vous' issues, that usually gets the ball rolling. Or ask them to read this.