Bitcoin and gold are having different functions, and therefore I don't know whether we can compare both of them. Bitcoin is a currency, which should be used for day-to-day expenses. On the other hand, gold is an investment asset, for which the main function is the protection of wealth. (That said, Bitcoin can be used for protecting the wealth as well, but it is it's secondary function).
Due to this volatililty , your statement is true. However on another note,
BTC could be an investment asset as well if there is a more stable price for the market also that in term of security , it is much more easier to secure your own
BTC than securing your gold .
So, if in future there is no any gold-mining more, I think the price will go higher and the gold handler can be rich immediately.
Basic law of Economic, yes only if there is a demand for gold otherwise no the price will still be the same
There are some factors that can affect btc price, and we don't know whether in future the price will raise or drop because of the factors.
Consider
BTC as a goods then you could apply economic law as well, Demands and supply. That would be the factor to consider (theorethically) however there are more things to consider as well, basically halving could be a factor that may help the price to raise or to plummet down
Due to lack of gold, governments might scrap it totally and make it worthless.
This is complete nonsense. Basically no one would do this as it serves no purposes . It is completely nonsense that someone would make something that valuable into a worthless piece of metal . Also with basic economics law, lower supply and higher demands will drive the price to be higher instead