If Bitcoin is not measured in dollars, what standard will we use to determine the price of Bitcoin? Even if this standard is acceptable in one country, it will not be acceptable in other countries. We need a unified global standard that is accepted by all countries of the world, and this will not happen unless the collective adoption of Bitcoin by all governments around the world.
This is theoretically, but in practice it is almost impossible, because there are many factors and variables that prevent the stability of the price of Bitcoin, and there are many people, you can even say the vast majority of them, no longer want or care about the stability of the price of Bitcoin.
It is highly unlikely that your statement will ever materialize, bitcoin assuming the position of fiat as a universal measuring unit is simply impossible. That will only happen if the government finds a way to control bitcoin and they still care about bitcoin.
When talking about bitcoin becoming a currency, everyone wants it to be used as a currency, but when talking about bitcoin's stability, no one wants that to happen. This is quite a controversial issue for any of us. We all want to maintain bitcoin's volatility and want it to become a currency, that will never happen.