As I read it, the stock price is basically modeled as a "true value" + noise. You can't observe the true value, but you can estimate it by characterizing the noise.
The effect of buying the stock with leverage instead of out of pocket (which is what call options let you do) is that the noise is amplified. The "true value" component doesn't change. Thus, when the noise is increasing the price, the price is higher than it otherwise would have been, and when the noise is decreasing the price, the price is lower than it otherwise would have been. Imagine the difference between P(x) = v(x) + cos n and P(x) = v(x) + 2 cos n.
Can technically move the market? Sure, but everything that happens in the market is technically moving the market. The main effect (as discussed in the Matt Levine pieces) is to increase the volatility of the stocks in question.
