You are right with a stable price everyone will rely on it for their money to put them on bitcoin, and they will not afraid that they will lose their money because of the volatility of the price, I think then all of the bigger companies will also adopt bitcoin for their businesses.
Indeed as people will see that its price is stable many will got interested with bitcoin. If they will see that it is just like the actual money, maybe they would also engage into more use of bitcoins. They would probably buy more bitcoins and invest it. Then more companies like what you said including stores and supermarket would include bitcoin as an option for mode of payments.
Yeah price stability will attract many new people to invest their money in bitcoin as they will be ensure that by investing in it they are not gonna loose their money and they will start treating it as an currency.