If so, then if company A never pays dividends to its shareholders, then what is the point of owning company A's stock?
The stock itself can go up in price. This is not necessarily pure speculation either, the company could just reinvest the profits and grow. Since you own part of a company, your share would also increase in value.
The company could also decide to start paying dividend. I think one rule of thumb is that growing companies won't pay out, since they reinvest all profit to grow even more, but very large companies like McDonalds or Microsoft who don't really have much room left to grow will pay dividends more.
Surely the right to one vote for company A's Board can't be that valuable.
Actually, Google for instance neither pays dividend nor do you get to vote. Basically all you get for your money is partial ownership of the company. This still gives you the right to seize Google assets if you go bankrupt, if there's any asset left once the creditors are done (credit gets priority over equity).
What is it that I'm missing?
What you are missing is that the entire concept of the dividend is an illusion. There's little qualitative difference between a stock that pays dividend, and a stock that doesn't.
If you were going to buy the stock, then hold it forever and collect dividend, you could get the same thing with a dividend-less stock by simply waiting for it to gain say 5% value, then sell 4.76% of your stock and call the cash your dividend. "But wait," you say, "that's not the same - my net worth has decreased!" Guess what, stocks that do pay dividend usually do drop in value right after the pay out, and they drop by about the relative value of the dividend as well.
Likewise, you could take a stock that does pay dividend, and make it look exactly like a non-paying stock by simply taking every dividend you get and buying more of the same stock with it.
So from this simplistic point of view, it is irrelevant whether the stock itself pays dividend or not. There is always the same decision of whether to cut the goose or let it lay a few more eggs that every shareholder has to make it. Paying a dividend is essentially providing a different default choice, but makes little difference with regards to your choices.
There is however more to it than simple return on investment arithmetic: As I said, the alternative to paying dividend is reinvesting profits back into the enterprise. If the company decided to pay out dividend, that means they think all the best investing is done, and they don't really have a particularly good idea for what to do with the extra money. Conversely, not paying is like management telling the shareholders, "no we're not done, we're still building our business!". So it can be a way of judging whether the company is concentrating on generating profit or growing itself. Needless to say the, the market is wild and unpredictable and not everyone obeys such assumptions. Furthermore, as I said, you can effectively overrule the decision by increasing or decreasing your position, regardless of whether they have decided to pay dividend to begin with.
Lastly, there may be some subtle differences with regards to things like how the income is taxed and so on. These don't really have much to do with the market itself, but the bureaucracy tacked onto the market.