Competition is for losers
if you're a winner. The problem is that winning requires beating all the losers.
The context of Peter Thiel's quote is that the really profitable companies are all ones where there are no real substitutes in their problem domain. But that's looking at the situation after-the-fact. In the process of getting there, they usually had to enter a very crowded market with a unique take on the problem; over time, other people realized their take was better, and so when the market consolidated, they were the last one left standing. Google entered a crowded search market (Altavista, Excite, Infoseek, Hotbot, etc.), Facebook entered a crowded social networking market (Livejournal, Xanga, EZBoard, MySpace, Friendster), Dropbox entered a crowded filesharing market (YouSendIt and dozens of clones).
If you read Steve Blank's books, one of his points is that it's basically impossible to take over a market where a single firm controls >74% of the market, and if a single firm has >26% of the market, you must spend 3x them in sales & marketing to have a chance. So the only markets that can be attacked by startups are the ones that are very competitive, or the ones that nobody knows exists. Of course, not all of those competitors are going to survive.
I suspect YC does select for companies that have a unique take on the market: "What do you know about the market that your competitors do not?" has been on the application form forever. They may not always get it right, but then, they don't have to: YC's unique take on the market is that they can fund hundreds of startups, take a small amount of equity, and still raise the expected value of your startup enough to be worth having as an investor.