[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.

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