Bitcoin will be more popular if the price is stable. More will choose bitcoin as an investment. Because bitcoin is a digital currency that is very easy to use.
How will you even earn from bitcoins as an investment when the price is stable? People buy/invest in bitcoin because they know that it's very profitable due to its volatility. The price is the one that everyone's looking at to see if their investment has made profit already or not.
If bitcoin had a stable price, I can only imagine the economy of it. How many bitcoins would there be? An unlimited amount? Supposed that bitcoins still gets recognition and popularity when it had a stable price instead, then that means people are getting bitcoins at a single source with a fixed price. Is that how things will be?