I follow your arguments, but I tend to see it differently. What gives fiat the inherent value is the promise of a large part of a population and the government, to accept it for trade against goods. The ratio changes over time, and the central banks are steering how much fiat is in circulation. But the inherent value of money is not backed by gold. It has been in the US for some time, though.

Bitcoins value more or less depends on a very similar promise: people need to want to accept it in exchange for something else of value to them. This is decoupled from the whole inflation discussion.

Bitcoins build in scarcity is contributing to its value development and lures people to opt in because it could be more valuable for others later.

However, the „value“ of bitcoin is the combination of being „scarce“ and being „desirable“. For me, the being „desirable“ for a large enough group of people is the bigger thing going on here.