I 100% agree with you on this becoming relevant before 2140. @AncapHodl made a great point about this.

You're also right that the fiat price can't double every halving forever. This is my pet theory but I think the market cap will rise until it reaches a significant portion of the world's economic value, then it will stop increasing exponentially and just grow proportionally with the economy.

However, I don't think that the price needs to double every halving in order to maintain or increase the hash rate. It may not even be the case that it must outpace the economy to be worth mining. That's like saying the price of gold needs to grow exponentially in order to continue to be worth mining. The purchasing power simply needs to be enough to pay for the miner's expenses with at least a little profit left over.