Here's what I learned:
The new SD card holder does NOT lock the card in place the way the previous one did. I will have to glue my cards in to make sure it doesn't wriggle out over time.
The issue with the SD card not being recognized was due the Arduino not recognizing a SD card was inserted. This is triggered by the little metal strips visible in the cut-out section of the SD card holder. The inner most of these strips was not well formed. I tried to bend it into a different shape but it is too flimsy, so I resorted to soldering it into place. Now the Arduino always thinks a SD card is present, but oh well.
In conclusion: the change to the SD card holder is a bad one. I wouldn't mind paying 1€ extra on a 50€ product to not have these kinds of problems.