~ Even if no more coins are mined, there will still be a supply. There will always be people who are willing to sell their bitcoins -- that's the definition of supply.
Furthermore, demand (and price) would not "instantly rock up". Scarcity does not affect demand. People won't want to buy more bitcoins just because they are no longer produced.
I agree, only for that reason they won't. Theoretically they could want to, if because of the hype, like it happened with CryptoKitties, having 1 BTC was in vogue, and the rich would buy it just to brag in front of their friends that they have one. I still don't discard that possibility completely, but I think that it is much more likely that the price of Bitcoin will be rising because of its utilization. If you want to pay for some goods and/or services with BTC, you should buy the currency first, right?