I think if Bitcoin was a stable price bitcoin would be dead right now. Because people invest in bitcoin because the value is not stable. They make money because of this and that is what people want. That is why everyone is here right now.
For starters Bitcoin is created as a currency in the first place and not as a mode of investment. Having a more stable price will of course be bad for profit takers or traders but on the consumer's end it will be good as businesses will now be more than willing to accept Bitcoin as a mode of payment because they know that the value of their earnings won't change or at least be near at what fiat currency has to offer. Bitcoin being a worldwide use currency will become a reality if that happens.
It was created to be a currency, but it has become a commodity. Nobody uses it as a currency (well, at least relatively compared to how many use it as a speculation vehicle), but it's been a smashing success as a gambling vehicle to allow people to speculate. As long as the price keeps going up and there's more and more people helping to fulfill the self-fulfilling prophecy, what's the worst that could happen? Well, that's exactly what the Tulip bubble was. And the tech bubble. And the housing bubble. See the problem? Bitcoin is on a parallel with all those disastrous bubbles where people deluded themselves into thinking that the price can't go down and you can only make money by buying it, just like tulips, dotcom stocks and houses before it.
After what bitcoin has become (a speculative asset/commodity), I'm not sure it can ever become a currency. To answer the the OP's question, nobody would be interested in it then.