Of course that would be possible. You could just say that one Bitcoin to mine costs a difficulty of x, and the difficulty rises (doubles) in a certain (block) time interval. But Bitcoin does not do it. (Maybe some Altcoin does.)

The hard limit of 21 million BTC does not really resemble the gold scarcity. Gold is not scarce because people have mined all the gold on earth. There is still a huge amount of gold, and mining levels are increasing, not decreasing. And people who believe that mineral prices can only go up should look what happened to the oil price recently. They are deluded by the peak oil myth. As prices go higher, miners will invest more money in innovation and capacity to increase mining rate.

For gold for example, most gold is in the inner core of the earth. Of course it is unrealistic for us to think about how to mine that, but it is not a hard limit like the 21M limit.

Gold production according to Wikipedia: