Ubuntu does work and is a decent distro in many ways. The problems are around how canonical leverages things for its own financial benefit for the detriment of users and the Linux community.
A good example is Snap. It is forced on users - even Firefox is a snap on Ubuntu. This is not an efficient way fo end users to run their system or their most used software.
Instead of making the builds available as standard software, users have to use the Snap or go hunting elsewhere for builds. That’s anti-user and is identical to how Microsoft behaves with windows. It doesn’t do things to benefit users, it does things to benefit Microsoft.
It’s arguable whether what snap does is actually worth the overhead - I can see that it is more secure in many ways. But then so it Flatpak, and that is more universally used for desktop software across Linux distros. Snap has some inherent benefits for server side use but then why force it on end users where it is not as good as Flatpak in many ways? Or Appimage?
So Ubuntu is fine in many ways, but why bother when you can go for alternatives and give the best of both worlds? Mint is an Ubuntu based distro without snap and other canonical elements. I used mint for ages, it’s great and there is a reason it’s so popular.
I’ve moved on to OpenSuSE now but the Ubuntu ecosystem is fine, it works well for many, and it’s very well documented and supported which often works downstream in Mint and others. It’s just Ubuntu itself thats a bit crappy due to the decisions made to suite canonical rather than what users want or would suit them best. In the end it all comes down to personal choice and what people are willing to accept from their distro.
Is there a reason you gave /var it’s own partition? Or is the problem that your entire root file system is full?
As others have said if you have a /var partition, resizing should fix the problem but the other solution would be to migrate the contents back to your main file system partition. Presumably at present there is a symbolic link folder pointing to your /var partition? Copy the /var partition contents into a new folder then boot in to recovery mode and delete the symbolic link and rename the new folder to /var. However presumably you have a good reason for splitting /var out.
If you don’t have a separate partition then the issue may be your root system itself is full and that partition needs resizing if possible or cleaning our to make space.
Finally, Flatpak does also use the /var directories in the home users folders (it uses this for single user installs of software vs system wide installs). It’s possible it’s axtually the home folder/partition that is full and that needs resizing or cleaning out to make space .