chore(build-and-push): add fail-fast SSH probe to remote builder host

Surfaces the real reachability/auth error instead of buildx's slow
'context deadline exceeded' when the runner can't reach the daemon.

build-and-push/action.yaml

@@ -64,12 +64,22 @@ runs:
       env:
         SSH_KEY: ${{ inputs.ssh_key }}
         BUILDER_HOST: ${{ inputs.docker_host }}
+        BUILDER_USER: ${{ inputs.docker_user }}
       run: |
         mkdir -p ~/.ssh
         echo "${SSH_KEY}" > ~/.ssh/id_ed25519
         chmod 600 ~/.ssh/id_ed25519
         ssh-keyscan "${BUILDER_HOST}" >> ~/.ssh/known_hosts
 
+        # Fail fast with a real diagnostic instead of a slow buildx
+        # "context deadline exceeded" if the runner can't reach / auth to the
+        # remote docker daemon over SSH.
+        echo "== probing ssh://${BUILDER_USER}@${BUILDER_HOST} =="
+        ssh -o BatchMode=yes -o ConnectTimeout=10 \
+          -i ~/.ssh/id_ed25519 \
+          "${BUILDER_USER}@${BUILDER_HOST}" \
+          'docker version --format "server={{.Server.Version}}"'
+
     - name: Set up remote Docker Buildx builder
       shell: bash
       env: