Re-land: Enable lints library_private_types_in_public_api, sort_child_properties_last and use_key_in_widget_constructors (#5692)

Re-lands flutter/plugins#5428

This is a revert of flutter/plugins#5691 (the revert of the above) with the following changes:
- Excludes the repo tooling changes that had to be added to the revert, since we want those
- Fixes local_auth:
    - Updates code for the new analysis failure
    - Fixes the bad version merge that dropped the version change
- Reverts a version change in `file_selector_platform_interface`, which didn't otherwise change

Author: stuartmorgan-g

script/tool/test/util.dart

@@ -319,14 +319,14 @@ String _pluginPlatformSection(
   return entry;
 }
 
-typedef _ErrorHandler = void Function(Error error);
+typedef ErrorHandler = void Function(Error error);
 
 /// Run the command [runner] with the given [args] and return
 /// what was printed.
 /// A custom [errorHandler] can be used to handle the runner error as desired without throwing.
 Future<List<String>> runCapturingPrint(
     CommandRunner<void> runner, List<String> args,
-    {_ErrorHandler? errorHandler}) async {
+    {ErrorHandler? errorHandler}) async {
   final List<String> prints = <String>[];
   final ZoneSpecification spec = ZoneSpecification(
     print: (_, __, ___, String message) {