Como solicitar que o usuário acesse o armazenamento do telefone?

Use o plug-in do manipulador de permissão do pub.dev, esta é uma resposta semelhante

para android adicione essas linhas no arquivo Androidmainfest.xml

<uses-permission android:name="android.permission.READ_INTERNAL_STORAGE"/>
<uses-permission android:name="android.permission.WRITE_INTERNAL_STORAGE"/>
<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE"/>
<uses-permissionandroid:name="android.permission.MANAGE_EXTERNAL_STORAGE"/>

para iOS, adicione o código abaixo no arquivo info.plist

<key>NSMicrophoneUsageDescription</key>
<string>This app require permission to access microphone</string>
<key>NSPhotoLibraryUsageDescription</key>
<string>This app require permission to access gallery</string>

modifique o código abaixo de acordo.

 Future<void> getStorageReadPermission() async {
    final storage = await Permission.storage.status;
    !storage.isGranted
        ? await Permission.storage.request()
        : debugPrint("Storage read permission already granted");
  }

  Future<void> getStorageManagePermission() async {
    final manageStorage = await Permission.manageExternalStorage.status;
    !manageStorage.isGranted
        ? await Permission.manageExternalStorage.request()
        : debugPrint("storage permission already granted");
  }

Finalmente descobri que perdi algo no meu código. E eu assisti o vídeo novamente e percebi.Tenho que colocar "if(permission.isAuth==true)" ao invés deste "if(permission==true)".

Em vez disso

import 'package:photo_manager/photo_manager.dart';

class MediaServices {
  Future loadAlbum(RequestType requestType) async {
    var permission = await PhotoManager.requestPermissionExtend();
    List<AssetPathEntity> albumList = [];
    if (permission == true) {
      albumList = await PhotoManager.getAssetPathList(
        type: requestType,
      );
    } else {
      PhotoManager.openSetting();
    }
    return albumList;
  }
  Future loadAssets(AssetPathEntity selectedAlbum) async {
  // ignore: deprecated_member_use
  List<AssetEntity> assetList = await selectedAlbum.getAssetListRange(
      // ignore: deprecated_member_use
      start: 0, end: selectedAlbum.assetCount);
      return assetList;
}

tenho que usar esse

import 'package:photo_manager/photo_manager.dart';

class MediaServices {
  Future loadAlbum(RequestType requestType) async {
    var permission = await PhotoManager.requestPermissionExtend();
    List<AssetPathEntity> albumList = [];
    if (permission.isAuth == true) {
      albumList = await PhotoManager.getAssetPathList(
        type: requestType,
      );
    } else {
      PhotoManager.openSetting();
    }
    return albumList;
  }
  Future loadAssets(AssetPathEntity selectedAlbum) async {
  // ignore: deprecated_member_use
  List<AssetEntity> assetList = await selectedAlbum.getAssetListRange(
      // ignore: deprecated_member_use
      start: 0, end: selectedAlbum.assetCount);
      return assetList;
}