UPDATE:

After a little more than 5000 miles, the GLC needed service (the 20.000 miles one), so we dialed the long term rental number and they told us to drive to the nearest SIXT location and we would get a replacement car. So we drove 500 miles to San Antonio, TX got there on a friday evening and unfortunately there was no car available in the same category we booked or higher. The man behind the desk said our best bet would be to drive to Dallas (which is another 400 miles away). Bear in mind that we only have 5.500 miles included in our rental agreement and after that we'll be charged 35 cents / mile + taxes. So the drive to San Antonio and then to Dallas would cost us more than $300 + taxes + gas. We told him we would stay the night in San Antonio and agreed to call back on Saturday to see if they had any availability the next day.

Still no availability the next day, but we were told we should definitely come back on Sunday, because they will definitely have availability then. Stayed another night in San Antonio and went to Sixt on Sunday morning where we were greeted by the German manager, and indeed, plenty of availability! We could choose between a Mercedes-Benz GLS, GLC, Cadillac XT5, BMX X3, ... The manager convinced us to take the X3 and what a great car it is!

On Monday we called the long-term rental number again (they are not reachable during the weekend), and let them know about the fact that we wasted 2 full days waiting for a replacement car. They told us they wouldn't charge us 2 rental days and would drop the extra charge for the additional miles we've driven to drive to San Antonio.
Very curious to see how many miles exactly they won't charge, as they can't possibly know how many miles the detour to San Antonio took. I think we'll drive an additional 5000 miles on top of the included ones, so the extra charge will be between $0 and $2000.