a disclaimer to start: neither gsmartcontrol nor smartctl will end up telling you that your drive is broken beyond repair. both can only run a couple of tests and show you any errors if they exist. the first photo you posted shows journalctl unable to write because your system was set to read-only which only happens when the file system is healthy enough to not just shut down, but only just since your system locks up and becomes unusable.
if you believe the problem might be ubuntu, you might consider trying a different distro. both fedora and opensuse are not based on ubuntu (and therefore debian either). if you want to try something familiar but upstream from ubuntu, there is always debian. it does look from here like the disk is not healthy though.
that being said, you can sometimes get gsmartcontrol to read a disk by going into the Device menu and selecting Add Device. you will need to know your device name (/dev/nvmeX) which you should be able to get from lsblk. you need the name of the disk (the first line), not the partition on the line below. for example, mine is /dev/sdb not /dev/sdb1. you put that in the “Device name:” field and add -d nvme to the “Smartctl parameters:” field. if that works, it would be helpful to look at the error log but it would be best to run both the short and long tests available under the Self-test tab. if the short test fails, the long test will most likely as well.
if gsmartcontrol still refuses, you may be able to run smartctl on the drive from the command line with:
sudo smartctl -a /dev/nvmeX.
again, you will need to get the actual device name to replace nvmeX. if that doesn’t work, you can try with
sudo smartctl -a /dev/nvmeX -d nmve.
if either of those works, the goal is to run both short and long tests with -t short and -t long. that command would look something like:
sudo smartctl -t short /dev/nvmeX.
the output will tell you how long the test will take. you would then need to run smartctl -a again to see the result. most short tests take about 2 minutes.
sudo smartctl -t long /dev/nvmeX for the long test. those vary depending on the size and type of disk.
i will repeat part of my disclaimer and add to what @Akito and the person on the bug report said: gsmartcontrol can only show you some numbers. at the end of all of this you still need to decide if you trust this disk with your information and time. the part about the warranty is a good thought and if the disk fails smart tests that may make it easier to file a claim.