Your simplified question isn't actually a question. But if it were, my answer would be "no". There are many ways in which that could not be true.
For example, as I already mentioned, public food safety ratings benefit new market entrants more than they do existing restaurants. Ergo, every existing restaurateur might oppose them (or at least decline to pay for them).
Another possible reason: dominant players benefit the least from public health ratings (because they already have a large established clientele and good word of mouth), but they are the ones with the largest surplus for things like this. So it might not happen because dominant players wouldn't work to make it happen. They might even oppose it.
Or another: humans have many known cognitive biases, including ones toward competition and zero-sum thinking. Perhaps prominent restaurateurs would oppose regulations that would benefit everybody because they would focus on benefit to competitors.
Or another: restaurateurs are very busy people, or perhaps very focused people, and so wouldn't have sufficient time or awareness to understand the benefits of a scheme like this.
Or another: even if they actually stand to benefit on average, individual restaurant owners might fear exposure of health issues, and would rather keep things quiet just in case (the loss aversion bias).
Or another: the costs of the program are short-term and obvious, but the benefits of the program appear speculative and/or long term, and nobody is willing to take the risk or take the short-term loss.
Or another: the program might not pay for itself purely in terms of increased restaurant business profits. However, it does pay for itself when health costs and human suffering are taken into account. In other worse, restaurants wouldn't do it because making people sick is a negative externality.
Or another: maybe nobody has gotten around to it yet. There are plenty of things that businesses do now that they didn't do at one point. Not because market conditions were different, but because things take time. Actual humans have to actually think of the idea and then get a lot of other actual humans to make it happen.
So off the top of my head, that's, what, eight holes in your argument from orthodox free-market dogma. Presumably an economist with relevant experience could find more, as could a public health expert or a restaurant owner.
Markets are a fine technology for accomplishing particular purposes. But like any real technology, they have practical limits and known failure modes. Which should be unsurprising given that they are implemented on people, who are rational to about the same extent that cows are spherical. [1]
[1] http://en.wikipedia.org/wiki/Spherical_cow