With M-: or M-x eval-expression. This was just to check what element Org would return for [cite:@key]. So when point is on a link, (org-element-context) returns a link element - a list starting with (link ...). If it’s on a headline - then a headline element (headline ...). I expected (after inspecting oc.el and relevant parts of org.el) that for a citation [cite:@key] this function would return (citation ...) but it returns (paragraph ...) meaning that Org parser did not recognize the citation as Org element and treated it as a part of a paragraph. This explains why following or exporting a citation does not work. But I can’t figure why Org element does not parse the citation in my instance. I therefore wondered may be you had any similar issues.
Actually, when I ran make in the cloned repository, the command quit with an error during building the Org manual. Lisp files were successfully compiled before that, so I assumed it was fine to continue. But maybe something else was not built properly and this was the cause of my issue.