If option prices are publicly quoted on an options exchange, why do people use Black-Sholes to estimate their price?

Part of the reason is that OTC (over the counter) options aren't publicly quoted (obviously). Another part is that the implied volatility on the option is not a measurable input to the BSM model but is the volatility used in hedging a portfolio by minimizing volatility.

The market price tells you what price you can currently buy/sell based on supply and demand. The Black-Scholes model is one way of estimating the fair market value. If they don't agree, then you may conclude that it's a good time to buy or sell.