[CP-stable] Split Linux docs_publish into Linux docs_generate_release (#169912)

This pull request is created by [automatic cherry pick workflow](https://github.com/flutter/flutter/blob/main/docs/releases/Flutter-Cherrypick-Process.md#automatically-creates-a-cherry-pick-request)
Please fill in the form below, and a flutter domain expert will evaluate this cherry pick request.

### Issue Link:
What is the link to the issue this cherry-pick is addressing?

https://github.com/flutter/flutter/issues/168913

### Changelog Description:
Explain this cherry pick in one line that is accessible to most Flutter developers. See [best practices](https://github.com/flutter/flutter/blob/main/docs/releases/Hotfix-Documentation-Best-Practices.md) for examples

N/A - This is an internal infrastructure change.

### Impact Description:
What is the impact (ex. visual jank on Samsung phones, app crash, cannot ship an iOS app)? Does it impact development (ex. flutter doctor crashes when Android Studio is installed), or the shipping production app (the app crashes on launch)

No user impact, could negatively impact how API docs are published.

### Workaround:
Is there a workaround for this issue?

N/A

### Risk:
What is the risk level of this cherry-pick?

### Test Coverage:
Are you confident that your fix is well-tested by automated tests?

### Validation Steps:
What are the steps to validate that this fix works?

View post-submits when a stable release is published and verify API docs were published.

.ci.yaml

@@ -586,18 +586,15 @@ targets:
   # to GCS", and conditionally on the "master" channel will also deploy the docs
   # to Firebase (https://main-api.flutter.dev/).
   #
-  # See "Linux docs_deploy_stable" for how the docs are deployed to stable API.
+  # See "Linux docs_generate_release" for how docs are built (but not published)
+  # and "Linux docs_deploy_stable" for how docs deployed for the stable branch
+  # (we do not deploy docs for beta).
   - name: Linux docs_publish
     recipe: flutter/docs
     presubmit: false
     backfill: false
-    # This means "allow this to be scheduled by the release/release_builder"
-    # recipe. Normally we'd use "schedule: true", but that *also* means "do not
-    # run this in normal presubmit/postsubmit", and we do want it to run in
-    # postsubmit for the "master" channel. Sorry.
-    #
-    # See https://github.com/flutter/flutter/issues/168709 for details.
-    schedule_during_release_override: true
+    enabled_branches:
+      - master
     timeout: 60
     dimensions:
       os: "Linux"
@@ -617,6 +614,67 @@ targets:
     drone_dimensions:
       - os=Linux
 
+  - name: Linux docs_generate_release
+    recipe: flutter/docs
+    # TODO(matanlurey): This has no effect, is used to allow creating a builder.
+    # Remove in the next PR.
+    # https://github.com/flutter/flutter/issues/168913
+    bringup: true
+    scheduler: release
+    presubmit: false
+    postsubmit: false
+    enabled_branches:
+      - flutter-\d+\.\d+-candidate\.\d+
+    timeout: 60
+    dimensions:
+      os: "Linux"
+    properties:
+      cores: "32"
+      dependencies: >-
+        [
+          {"dependency": "dashing", "version": "0.4.0"},
+          {"dependency": "firebase", "version": "v11.0.1"}
+        ]
+      tags: >
+        ["framework", "hostonly", "linux"]
+      validation: docs
+      validation_name: Docs
+      # TODO(matanlurey): Neither of these properties are actually used, since
+      # the branch name is not "master", but if they are removed, the recipe
+      # will fail. See https://github.com/flutter/flutter/issues/169108.
+      firebase_project: main-docs-flutter-prod
+      release_ref: refs/heads/master
+    drone_dimensions:
+      - os=Linux
+
+
+  # This step runs on the release channel "stable", after the same commit SHA
+  # has been run and built by Linux flutter_release_builder as part of a release
+  # candidate branch (i.e. /flutter-\d+\.\d+-candidate\.\d+/) in the previous
+  # target, "Linux docs_generate_release".
+  - name: Linux docs_deploy_stable
+    recipe: flutter/docs
+    scheduler: release
+    presubmit: false
+    postsubmit: false
+    enabled_branches:
+      - stable
+    timeout: 60
+    properties:
+      cores: "32"
+      dependencies: >-
+        [
+          {"dependency": "dashing", "version": "0.4.0"},
+          {"dependency": "firebase", "version": "v11.0.1"}
+        ]
+      tags: >
+        ["framework", "hostonly", "linux"]
+      validation: docs_deploy
+      validation_name: Docs_deploy
+      firebase_project: docs-flutter-dev
+    drone_dimensions:
+      - os=Linux
+
   - name: Linux docs_test
     recipe: flutter/flutter_drone
     timeout: 90 # https://github.com/flutter/flutter/issues/120901
@@ -6924,30 +6982,3 @@ targets:
         ["framework", "hostonly", "shard", "windows"]
     drone_dimensions:
       - os=Windows
-
-  # This step runs on the release channel "stable", after the same commit SHA
-  # has been run and built by Linux flutter_release_builder as part of a release
-  # candidate branch (i.e. /flutter-\d+\.\d+-candidate\.\d+/) in the previous
-  # target, "Linux docs_publish".
-  - name: Linux docs_deploy_stable
-    recipe: flutter/docs
-    scheduler: release
-    bringup: true
-    enabled_branches:
-      - stable
-    presubmit: false
-    timeout: 60
-    properties:
-      cores: "32"
-      dependencies: >-
-        [
-          {"dependency": "dashing", "version": "0.4.0"},
-          {"dependency": "firebase", "version": "v11.0.1"}
-        ]
-      tags: >
-        ["framework", "hostonly", "linux"]
-      validation: docs_deploy
-      validation_name: Docs_deploy
-      firebase_project: docs-flutter-dev
-    drone_dimensions:
-      - os=Linux

TESTOWNERS

@@ -304,8 +304,8 @@
 /dev/devicelab/bin/tasks/android_display_cutout.dart @reidbaker @flutter/android
 
 ## Host only framework tests
-# Linux docs_deploy_beta
 # Linux docs_deploy_stable
+# Linux docs_generate_release
 # Linux docs_publish
 /dev/bots/docs.sh @Piinks @flutter/framework
 # Linux packages_autoroller