That's a far more limited claim. There's a very common claim that "markets are efficient". Extending to all markets all the time. Explaining why market based solutions are the best way to provide health care and education and other (arguably common) goods.

But many markets (the labor market, the health care market, etc) work in such a way that it's ridiculous to assume that most participants know all public and private information all or most of the time.

So called "Perfect Information", as you point out, is an important requirement of the efficient market hypothesis, but not the only one. It also requires fungible goods (e.g., if I start making bicycles that are better and cheaper than your bicycles, they will necessarily sell more. That's often not the case, especially if there is strong branding).

Another requirement is that entrepreneurs have fair market access to capital. This assumption is true in many developed countries, but not in many developing countries where you need connections to obtain a loan. But even in developed countries, it's difficult to get extremely large loans on the merits of the financials alone, which explains why large infrastructure projects like highways, airports, and power-plants often don't get built solely by the private sector.

Private information here refers to whatever insights you bring with your research. The point is that if you gain some private information (by being a quant, analyst, etc.) you would immediately trade on it and the price would then start reflecting your views.

Also EMH, and that's the point Fama uses a lot to battle critics, does not put a time frame on when the price adjustment happens.

In a truly efficient market, you wouldn't need to haggle about the price. You would have a slew of options that you could choose from, and would choose the option that best fits your demand. In an efficient market, if you were ever being overcharged by a company, another company would pop up overnight, and you would immediately switch from one to the other. No haggling required, just knowledge of what's available and easy movement from one option to another.