And I think the only reason stocks and other standard exponentially-increasing investment vehicles are exponentially increasing, is because they entitle you to a constant fraction of the exponentially-increasingly-valuable economy.

Exponential increase is a consequence of the fact you don't have enough capital to exhaust investment opportunities or affect the market (much). Suppose I can buy G for £100, which will give me £110 at the end of the year. Then I can buy 2G, or 3G, or xG, giving me £x110 at the end of the year. So my profit is always proportional to my capital (as long as my capital isn't too large), so exponential growth of investment is the natural state of being.

Add uncertainty, and you get the random walk on a log scale (that's a little bit more subtle, and needs some other assumptions, but the fundamental argument is similar).