Test Insights (including flaky tests) is keyed to a specific branch. The All Branches view does not show Test Insights. The empty state tells you to select an individual branch. That is expected. It is not caused by GitHub App versus GitHub OAuth, and the branch id in the URL for All Branches is not a git branch.
Git tags are not branches. Pipelines started by a tag do not appear in Insights graphs or in the Insights API. Moving tests and deploys off a branch onto tags removes the only branch Test Insights can use, so flaky-test Insights goes empty even though the tag pipelines succeed.
The Insights API has the same tag exclusion. See Insights API endpoints not returning data.
Keep Insights while deploying from tags
Run a branch-triggered test workflow (for example on main) so Insights has a branch to select. Use tags for deploys.
Individual job Tests tabs still show results for tag runs when those jobs use store_test_results. Those tabs are per job. They are not the project Test Insights page.
Project-level test data for a branch you still build is available from the Insights API for that branch. Tag pipelines stay outside that API.