Alright.
A quick comment on the different ways you posted:
echo $page->uri();
You already wrote that, but: This returns the URI for the current language.
echo $page->uri('sv');
I just tested this syntax in the langkit on a project page and got the expected output. So that’s quite strange. Could you please verify if the same code works directly inside the template of your site?
echo $page->content('sv')->get('uri');
This is not supposed to work at all. The URI is not part of the content. The prior syntax should work.