I understand the concept of liquidity and the reward mechanism you describe but it doesn't ultimately answer my question, which why is the process of buying an existing share called "capital allocation"? Let's say I buy a GOOG share from Larry Page. Is it the idea that I "allocated capital" to Larry Page's bank account? It seems to me like the correct thing to say would be that I provided liquidity to Larry, not that I allocated capital. Or is it the idea that I allocated some of my own capital to the stock market?
Two points, one going to your question and one is of general interest.
Buying existing shares is "capital allocation" with respect to the _buyer's capital_. So I might allocate 20% of my capital (ie, gross financial worth) to being in shares of some company. The seller is allocating their capital somewhere other than the share. So you allocate your capital to GOOG, pay Larry and notify Alphabet that you are one of their capitalist overlords. If the price of Alphabet stock goes up, you now have more capital even though if you do a quick count you'll discover you have no new currency/cash. The reason this is important is that the people with a good ability to allocate capital will end up with more capital hence control. Eventually, the people in charge will be the people with a good grasp of what is changing (which I'll claim is desirable with no support).
For general interest, I've no insight into the intricacies of the US system, but in Australia every so often a company creates and sells new shares directly on the market. The upshot of this is a company can access the market directly for capital.
