The claim is not about "competition not working", the claim is about the price. It's actually quite tricky to determine whether the price level corresponds to monopolistic or marginal cost price, because it depends (in neoclassical theory, at least) on supply and demand and these are often not known. So people can easily mistake the (actual) monopolistic price for the (expected by neoclassicals) price at marginal cost, and I would argue they often do.

And there are other mechanisms, in play, too. For example, if there is friction in the market, and it often is, then the prices can be lower due to competition for market share.