Is the existence of equity trading a sign of an inefficient market?

In this case I am referring to market efficiency in the sense of supply/demand equilibrium due to competition. If I understand correctly, in a maximally efficient market, the intention is that prices shouldn't be able to remain artificially high for long because competing businesses can undercut them, and wages shouldn't be able to remain artificially low for long because competing businesses can offer better wages to attract employees.

This, of course, relies on their being a sufficient amount of competition. Ideally, participants in the market who possess enough capital are incentivized to invest in the formation of new businesses in order to make profit. But there is the issue of anti-competitive offers where larger companies offer to buy and either consume or bury new competition. If I understand correctly, the mere existence of such practices doesn't necessarily prevent the market from tending towards competition, because there will always be people with a long-term mindset who refuse to sell. That is, the market doesn't need most new businesses to be willing to refuse anti-competitive offers; just enough of them to make it inevitable that competition will eventually arise.

Now the main question. Larger companies also have a second trick up their sleeve to merge or bury competition even when the current owners of the competing company have a long-term mindset that leads them not to sell: equity offers. Basically offers to buy the competing company in exchange for shares of its equity once it is assimilated into ownership of the larger company, rather then just a one-time lump-sum sell.

Since larger companies have more advantage in economy of scale, and because decreased competition makes an industry more profitable, it is pretty much guaranteed that the dividends from such equity would be worth more to the owners of the smaller company than the return on investment of continuing to operate the company separately and competitively. That means that even if the current owners are refusing a lump-sum sell for the sake of the long-term profit, there is basically no reason for them to turn down a sufficiently high equity offer. And if such a sell occurs, since, again, less competition increases the profitability of the whole industry, the former owners would now be incentivized to avoid using their stake to make decisions that would be competitive against the original larger company.

Am I missing something? Because if I'm understanding this dynamic correctly, it would seem that, as long as equity trading is possible (which it must be in a free market, even if no public stock market existed, because of private contract equity), markets don't tend towards more competition, but rather less. Are there other factors I'm failing to consider?