A classic example (presented to my MBA class) was the two adjacent pizza parlors in NYC competing on price and driving the selling price of a pizza down to less than a dollar, whilst a shop a block away and around a corner was able to keep normal pricing.
but you also compete on price in the marketplace--not so much as to ignore your costs, as you note, since negative margins generally don't lead to viable businesses, but price competition nonetheless.
your example is the simplest game-theoretic version of price competition, which is more of a teaching model than a practical application.
(dynamic) price discrimination and other marketing tactics are typically employed to ensure positive margins even as you compete on price in the marketplace (e.g., airline seats).
no. business school teaches you that you can compete on price (cost strategy), or on value (differentiation strategy).
if you compete on price, you’re betting that you are, or will be, the most efficient provider in the market (e.g., walmart and its supply chain dominance). it’s completely viable/acceptable to compete on price. what you don’t ever want to do is price on cost, rather than on value provided.
you can also compete on being better in many other dimensions, which is lumped into the broad differentiation strategy category. you are then betting that you are better on your dimension and that customers really value that dimension (more than price).