Singleton Database wrapper
Singleton

I wouldn't call your Database class a singleton wrapper, as you can have multiple instances of it (not of the PDOObject field, but of the Database class). 

See here and here for actual singleton pattern.

I don't think that your approach is necessarily bad though.

Is this overkill?

I think that you are a bit dishonest in your example. Your not actually using BindParam in your code, and you would still have to catch the exception that is thrown. So instead of saving multiple lines per query as you suggest, you are actually only saving one, which isn't that much.

But doing it this way does provide you the ability to easily add functionality to executing a query, or change how you execute queries all together. I think that this is a big advantage and a good case for having this class.

Returning in the constructor


  return self::$PDOObject;


Why are you returning the PDO instance in the constructor? This seems a little odd.

Style

I say this a lot, but good style actually is important and leads to easier to read code.


remove all unnecessary newlines. You have a lot of newlines, which makes your code hard to read
be consistent with your curly brackets and indentation (always indent the same amount, always have curly brackets close at the same indentation as the opening statement).  
also be consistent with spaces (try{ -> try {)
comments: //handle exception isn't a useful comment, especially if it is handled by just throwing the exception :)