Thank you all! Definitely some great info here :-)
I'll go looking for a couple of nice C/S oval head slotted screws.
At first I thought that the screws pictured would not fit, but it turns out thet the holes are around 9mm deep. So that looks right. There also seems to be some remnants of the course thread left, which means that these self tapping screws were once fitted (which I didn't doubt).
On the topic of McMaster Carr: I have some experience there as I managed to order almost $70 worth of nuts and bolts, including 400 sets of 6-32 nuts, bolts and washers (impossible to get here, and definitely not that cheap) for my side curtain restoration project.
I had them shipped to an acquaintance in the US. He loaded them in a container that was already coming to this side of the pond. Saved me a lot in shipping costs (The package was heavy).
I don't think that I have to do this again, but we'll see
