If the price is fixed, then why would people invest their money in Bitcoin? Bitcoin is considered a store of value, the price will increase. If it becomes fixed, then the appreciation that users desire will stop, and no one will put their money in. Similarly, if the price is fixed, there is also no need to store it for future generations. In the beginning, users stored BTC with the hope that its price would rise over time. If it is fixed, then children will only sell it at that price.
The reason why people are rushing into investing in Bitcoin is because the price is increasing and the could get some profit from it when ever there is bull and this seems to be the most interesting part of Bitcoin.
If it price become fixed people will the buy Bitcoin but it won't be at the rate it is today depending on individual, and purpose of investing, I am sure at that point many will buy as a store of value and not for profit making. It may also drop the decentralization purpose of Bitcoin, which means the authority controlling it will have to decide when to sell, how to sell, what price to sell, and who to sell to. This will disqualify the original purpose of Bitcoin and makes no difference from fiat.