Put another way: price is determined by need and supply (aka, demand curve meets supply curve).
I would pay anything for air if I needed it, but I will gladly sell air in my yard for $1/m^3 because that air is worthless to me.
Is air priceless or worthless?
That is why price != value as most people think of it.
Not disagreeing with you, but isn't that already obvious from the fact that economic activity happens in the first place?
If you buy 5 apples from me for $5 then two things must be true: 1. The value that those 5 apples have to you exceeds the value that $5 have to you, at least at this very moment. Otherwise you would hang on to your $5 instead. 2. The value that those 5 apples have to me is less than $5 have to me, otherwise I would hang on to the apples.
The price of those 5 apples at this moment may be $5 but that doesn't reflect the value they have to neither me nor you. It's not the avereage either, necesarily. The only thing we know is that the value of them to you is higher and to me is lower.