Is the price really giving it value? Price is just a representation of value. It doesn't give it and it doesn't take it away.

Scarcity by itself doesn't give any value to things. You could see it in play when altcoin developers tried to make coins even more scarce than bitcoin and with even higher price per coin or token.

Some of the most important factors that give value to bitcoin are utility and uniqueness. It was the first to do it, the first idea of this kind of private money that was at the same time fast safe and secure. It caught people's attention and there you have it. The rest is history.