At in the  private SQLiteDatabase getDatabaseLocked(boolean writable) method, the path to the database is
 final File filePath = mContext.getDatabasePath(mName);

In your version the path is mName when you try to open read/write and
String path = this.mContext.getDatabasePath(this.mName).getPath();
if the first try does not work.

The effect is if the working code with Android sqlite classes is modified to use your  version of sqlite classes, the code will not work because the database cannot be opened.