In dealing with your question I will correct some misconceptions regarding the context.
doing a lot of I/O on the SD card (and not on an USB stick as it is recommended)
This is not really true; there is nothing wrong with using the SD card and I'd bet that the significant majority of pis out there do run the root fs on the SD card.
Point being, simply using the SD card for the root filesystem is not something that will inevitably lead to problems, and simply booting from USB is not something that will inevitably solve them.
No other LED is flashing next to it (which should be the WLAN, if I am correct)
There's no WLAN light. By default the green "ACT" led represents I/O activity on the SD card. If the card is formatted correctly but some of the required content (bootloader firmware, OS kernel, etc.) is missing or corrupted, the green led may flash in a steady rhythm indicating the problem.
But that is not happening in this case. In a healthy Pi no green led is what you get if you plug it in with no SD card or an SD card which is completely unreadable for some reason (eg. improper or corrupted partition table).1
On a "non healthy" Pi it may mean the SD card reader is wrecked, or the holder is loose (sometimes may work if you fiddle with the card).
A connected monitor does not get a signal.
The LEDs next to a connected Ethernet cable do no start to blink as I know it >
from other Ethernet devices. The cable is connected to a properly working router.
This is consistent with it not booting at all. Again: You will get exactly the same result with no SD card.
How would I have to mount the SD card to make it available under Ubuntu?
It is just a normal, DOS-MBR formatted card with a vfat and ext4 partition on it. It should be easily readable on any linux system the same way you would read another another normal SD card.
If normal mounting fails, this is good in your case, because it implies the SD card is the problem. If you have created a card fresh from an image and it fails and it also won't mount normally in a ubuntu system then the card was not created properly or is physically defective.
You should be able to run
fdisk -l on the card in ubuntu even if it does not mount properly. A valid Pi OS card will look something like this:
Disk /dev/mmcblk0: 29.7 GiB, 31914983424 bytes, 62333952 sectors
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disklabel type: dos
Disk identifier: 0x2292ede0
Device Boot Start End Sectors Size Id Type
/dev/mmcblk0p1 8192 532479 524288 256M c W95 FAT32 (LBA)
/dev/mmcblk0p2 532480 62333951 61801472 29.5G 83 Linux
If it is somehow drastically different -- such as failing to read a partition table at all -- then the card is no good either because it was created incorrectly or is physically defective.
If you can read the card OK, then have a look at the end of
/var/log/messages, these may have clues about the last thing that happened when the log was writable. Unfortunately, based on your description I do not think the system, if it started at all, would have had the opportunity to save any information.
- On some models with certain firmware it may come on solid (unblinking) instead.