Hi,
good to see that this fixed the styling!
Now, it looks like the Extension is helping a little bit, as it flips the table two two columns. If that's ok for you, you can of course continue using the Extension.
Now, otherwise, this is more of a general problem with tables in browsers. There's just no fixed responsive behavior that works for tables. So "a responsive table for any purpose" is not really a TablePress issue, but an issue for tables in browsers, but actually even more like "What is a responsive table, anyway? What behavior should it be showing?"
It's rather complicated from a philisophical point of view, I guess :-)
Now, there's another issue on your site: For some reason, I'm seeing the "Edit" screen below the table, which I should not be seeing, as only logged-in users can see that. However, clicking on it brings a 404 Not found error, which means that you are probably hiding your wp-admin folder somehow (which is fine). However, as I can see the link, there's something wrong somewhere with how pages are generated. Are those cached somehow, so that maybe the version that a logged-in users sees is shown?
Regards,
Tobias