It sounds like a problem with the SD card. I would recommend running chkdsk /r with the SD card. You should also make sure the data partition is formatted as FAT32 if it isn't already.
After the chkdsk, you might need to reformat the SD card and start again.
Reporting back as promised. I checked the card and found no errors. Tried again with another card - got exactly the same result: Just moving the cursor between the different menus on the home screen, everything seems fine. But 10-30 seconds after entering one of the menu items, the Switch freezez completely. I noticed one thing, however - everytime I went into the Album, the homebrew apps weren't showing up correctly with icons. Not knowing what to do (and feeling quite confident I did everything correctly, I also put ChoiudujourNX on my card, hoping that somehow I could update my CFW to 9.1.0). Strangely, when I rebooted, the homebrew icons were all in place, and I succesfully updated my emummc to 9.1.0. And ever since then, it has been running flawless. Everytime I booted into one of the failed attempts I never saw the Sept-bootlogo before the Atmosphere logo, but I do now. As I did everything the same every time (except also putting DNX on the SD card for the final and working try), my own guess is that maybe Atmosphere isn't quite optimized for use with an old software version (my Switch is 2.3.0) - but obviously I could be very wrong. Don't know if this helps anybody, but I at least wanted to let everyone know how I faired.