Good discussion - adding one thing that I didn't see mentioned re:
Also, after the initial influx of cash once a company goes public, where is the ongoing benefit for the company?
IANAMBA* but here's my naïve take:
A company (or rather a company's management) cares about the share price because
that's its job.
To put it less glibly: Under most company charters, shareholders own the company, and the management of said company (from the CEO on down) is there to serve
them. A CEO is hired by the board (aka the shareholder's elected representatives) and is incentivized with performance bonuses, and ultimately, with threat of termination. The CEO has a very real desire, then, to see that the share price rises (or that shareholders receive increased income via dividends/buybacks). This motivation (ideally) propagates down the chain of command: employees are rewarded/penalized according to how well they serve the goals laid out by their managers.
This is more than just theoretical. Activist shareholders (e.g.
Carl Icahn) can and do make substantial investments in companies with the express purpose of overthrowing management when they believe it can be run more efficiently. If you own an individual share of a company, you may wield essentially no direct power, but the voting rights (combined with the rights to future dividend streams) still give that share value.
* I am not an MBA.