That's not Peter Thiel's argument. That's Peter Thiel delivering a remedial microeconomics lesson. 'Perfect competition', 'monopolistic competition' and 'monopoly' are technical terms which don't necessarily mean what they might appear to. In perfect competition, for example, all goods are fungible. It's a good model for the commodities market, since one barrel of oil is very much like another (there are different grades of oil with different sub-markets, such as 'Brent crude' or 'west Texas intermediate', but you don't care which particular company extracted which particular barrel of oil from the ground within a given market).
Now once you introduce the element of skill, you have something more like monopolistic competition. Pizzerias are in competition, and they're selling a similar product, but they can compete on everything from the quality of the pizza dough to the comfiness of the seating, so different customers may end up strongly preferring one to the other. Maybe I'm price-sensitive and prefer Mario's bargain slice, but you're a gourmet and you prefer Luigi's home-made flavor. Someone else prefers the fancy service at Gino's, and so on. This is why towns can have so many different restaurants and why supermarkets can sell multiple different brands of what are essentially similar products: you can differentiate products and services, and anchor consumers to a particular combination.
Sure a new business may add value by providing a better service than the competitor it beats, but the overall wealth pie of this nation doesn't increase.
Oh yes it does, because now the wealth is being allocated more efficiently. Commodities are one-size-fits-all, but relatively few goods are commodities. Consider, for example, that there are relatively few companies in the business of wheat production (commodity) but a great many different bread companies, in line with the wide variety of different bread products.
Please, pick up a microeconomics textbook. Macroeconomics is so flawed and obtuse that a lot of people dismiss the musings of economists, but microeconomics is a very robust discipline that is almost criminally underappreciated. If I could back to high school and give my teenage self some advice, it would be 'learn microeconomics.'