I have created a GitLab project with a wiki but no code repository. When I visit the project URL, the content of the wiki Home page is rendered (where the README is normally rendered).
However, if I log out of GitLab and visit the project URL, the Home page is not rendered. Instead a list of recent issue activity appears. I can still visit the wiki while logged out.
How can I make the Home page visible at the project URL to logged-out users?
You cannot make the wiki Home page visible to logged-out users who visit the project page. The behavior you have described is as intended and cannot be configured.
Issue gitlab-org/gitlab-ce#12922 requests support for "wiki-only projects" and is open and unresolved in July 2018. If implemented, it would resolve the issue for your use-case (and require an updated answer here).