Did I get any of the facts wrong? Is my reasoning flawed?
you can pull the Ace and say that the original value of a bitcoin was equal to the electricity (electrons, copper wire degraded, silicon & chipsets depreciation) that went into the creation of it.
This is called the labor theory of value and it was subscribed to by early economists like adam smith. Today very few economists mainstream or otherwise accept the labor theory of value. It is generally understood by economists today that the value of an object is determined by the relative subjective valuations of consumers. Bitcoin was valuable early on not because of the labor required to make it nor the precedent of value but rather the foresight of some clever speculators who recognized the utility as a medium of exchange that it could have in the future and were willing to take an extremely high risk/high reward gamble.
In fact, MJGrae define the cost to produce a thing as its value. It is wrong. From this definition a hand made sword would be more valuable (and would cost more to buy) than the same sword produced in a factory at a lower cost. On the market, they are two swords with no detectable difference, so they will be valued the same from the buyers.
If the cheapest (in this case the industrial) producer is able to satisfy the demand enough to push the price to a level where the artisan can not make a profit (but he can), the artisan will exist the market.
Competition with other industrial producers (and other goods and services) will drive the price nearer the production cost.
This is the reason in the end prices approximate production costs.
Anon136 "speculators ... willing to take a... gamble" is on the point.
A speculator is any and all agents acting on the market, because they act now for a future reward be it a far or near future and be it a small or big reward.
So as the speculators take the gamble, they establish price levels and demand schedules. On these the market forces act moving the prices up and down in relation with supply and demand.