I had a similar issue. I was able to gain access to repos i couldn't see by going here:

https://github.com/apps/claude-for-github/installations/select_target

Using this link, select the github user or organization, the all or specific repos to delegate access to the claude-for-github app.

Once granted, you still may not see all repos in the list, but I was able to successfully use the "paste Github URL" link in the Claude desktop app to add a private repo that did not appear in the list.