tl;dr: There was a case whereby some team-owned repos in Bitbucket were not selectable when using the “Create a new site” flow. We fixed that, with your help.
When creating a new site and using Bitbucket as your Git provider, any team-owned repositories would not be listed for selection unless you were an administrator of the team - even if you are an administrator of that repo within the team.
The bit Bitbucket API provides different configuration options when we fetch the list of repos. To get the full picture, we make different requests with different options configured (one for the user, one for the team, etc). This particular scenario was falling down a crack between the options, and those repos described above were never being returned.
We’ve tweaked the options to make sure they’re covered, and all repos for which the user has the appropriate level of access should now be listed so they can be chosen as the source of a new site.
Thanks to the wonderful community here for reporting this very nuanced issue and enabling us to track it down and deploy a fix.