Make the Merge-Bot Task's delete-branch Input Delete Heads After an Auto-Merge #2296

Problem

merge-bot-task.yml's delete-branch input appends --delete-branch to gh pr merge --auto. With --auto, the merge happens later on GitHub's side, once the required checks pass, and nothing deletes the head branch at that point. So for a repo that keeps repository-wide auto-delete off, the input has no effect, and the bot heads it exists to clean up stay behind.

Evidence

ptr727/homeassistant-purpleair keeps delete_branch_on_merge: false. Its carried merge-bot ran gh pr merge --auto --delete-branch --squash on the bot PR ptr727/homeassistant-purpleair#358. The PR auto-merged at 2026-10-03T02:08:30Z. Afterwards, git ls-remote origin refs/heads/ha-version-bump/matrix still returned df3a7e1, which is #358's merged headRefOid.

The hub task builds the same argument list (args=(--auto "$method"), then +=(--delete-branch)), so every adopter setting delete-branch: true inherits the gap.

Docs that describe the input as deleting bot heads:

• docs/reusable-workflows.md, "Adopting the Merge-Bot"
• the input's own description
• the adopters' stub comments ("Bot heads are therefore deleted here explicitly")

Possible fixes

• Delete the head after the merge actually lands, for example from a pull_request_target: closed job gated on merged == true, the bot author and the in-repo head, verifying the remote tip against the PR's headRefOid first.
• Or drop the input and document that bot heads persist. A rolling branch like ha-version-bump/matrix is recreated by its tracker anyway, while Dependabot deletes its own branches.

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