Reland: "Reorganize and clarify API doc generator" (#132353) (#132710)

## Description

This re-lands #132353 with some additional options for keeping around the staging directory, so that the recipe for publishing docs can give those options and have the staging directory left around for deploying to the website.

Reverted in #132613

## Related Issues
 - https://flutter-review.googlesource.com/c/recipes/+/49580

dev/docs/dashing_postprocess.dart

@@ -1,29 +0,0 @@
-// Copyright 2014 The Flutter Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style license that can be
-// found in the LICENSE file.
-
-import 'dart:io';
-
-/// This changes the DocSetPlatformFamily key to be "dartlang" instead of the
-/// name of the package (usually "flutter").
-///
-/// This is so that the IntelliJ plugin for Dash will be able to go directly to
-/// the docs for a symbol from a keystroke. Without this, flutter isn't part
-/// of the list of package names it searches. After this, it finds the flutter
-/// docs because they're declared here to be part of the "dartlang" family of
-/// docs.
-///
-/// Dashing doesn't have a way to configure this, so we modify the Info.plist
-/// directly to make the change.
-void main(List<String> args) {
-  final File infoPlist = File('flutter.docset/Contents/Info.plist');
-  String contents = infoPlist.readAsStringSync();
-
-  // Since I didn't want to add the XML package as a dependency just for this,
-  // I just used a regular expression to make this simple change.
-  final RegExp findRe = RegExp(r'(\s*<key>DocSetPlatformFamily</key>\s*<string>)[^<]+(</string>)', multiLine: true);
-  contents = contents.replaceAllMapped(findRe, (Match match) {
-    return '${match.group(1)}dartlang${match.group(2)}';
-  });
-  infoPlist.writeAsStringSync(contents);
-}