It doesn't mean your game is broken simply because you don't have an item like Leftovers, if that were true then every Pokemon game I own is broken. You simply haven't found it yet or you're looking in the wrong places.
You can find Leftovers using the itemfinder on Routes 12 and 16, where the Snorlax you mentioned was sleeping. If you've already been looking in those places (which you should have specified if you were), remember:
- Make sure you have you itemfinder on. The Leftovers are different in this game to other game, they require you to use the Itemfinder to show up at all.
- The itemfinder will only tell you the Leftovers is nearby if you're standing on it, another thing unique to these games. If you stand on it you'll be able to get it.
So just put your itemfinder on and just keep searching.