According to that same reasoning you mentioned, could we permit eating fish with meat?
By the way, I looked for the parallel ruling in Arukh HaShulchan and found that he specifically agrees with the view of the Shulchan Arukh and forbids it:
Arukh HaShulchan, Orach Chaim section 4, section 15:
“And one should not touch with his hand before washing the mouth, nor the nose, nor the ears, nor the eyes.”

What did you mean when you mentioned Arukh HaShulchan, Yoreh De'ah?

As for fish, in principle yes. Although on that everyone really is careful, and it should be discussed from the standpoint of custom.
As for Arukh HaShulchan, I don’t have it accessible right now. I remember something there about uncovered water.

Regarding the fish,
There is, after all, a well-known rule that a custom practiced because people erred and thought that this is what the Jewish law is, when in fact that is not the Jewish law, may be permitted without formal annulment. And it is reasonable to assume that most of the Jewish people practiced this custom because they were mistaken and thought there was danger involved and that this was the Jewish law. So wouldn’t it be possible to permit it without annulment? It would be one thing if you said that the custom began as a stringency and an act of piety, like: “Rabbi Zeira said: The daughters of Israel were stringent with themselves, that even if they see a drop of blood the size of a mustard seed, they sit seven clean days because of it,” or like the evening prayer. But here that does not seem to be the case.

As for Arukh HaShulchan, I think this is what you meant:

Arukh HaShulchan, Yoreh De'ah section 60
Subsection 2
“But if it ate a human deadly poison, or a snake bit it, or a rabid dog bit it, and the like, it is permitted regarding treifah but forbidden because of danger to life. And as for why we are not concerned about crushing by a snake, we explained this in section 57, see there. And perhaps you will ask: why did we include uncovered water among the deadly poisons of an animal, for there is also danger for a person to drink uncovered water in places where snakes are common, as is well known throughout the Talmud? Certainly that is so. If we were to see any sign of harm in the animal after it drank the uncovered water, we would forbid it to a person because of danger to life, just as with human deadly poisons. But when we permit it, we are speaking of a case where we see no sign of harm in the animal after drinking it. And this is not like all the other cases we listed, where a sign of harm is seen, such as reddened blood and the rest, and nevertheless we permit it. But drinking uncovered water we only permitted when there is no sign of harm. And the reason we compared it to the deadly poisons of an animal, which are also permitted, is because certainly the snake did not drink from them, for had it drunk, some sign of harm would have been noticeable in the body of the animal [Taz, end of section 116].”

Regarding the fish, you’re right. That’s why I wrote only that it should be discussed in terms of custom, not that it is binding. The point to discuss is exactly the one you raised.
But why, in my opinion, is it still worth discussing? Because even if in my opinion there is no danger here, and even if in my opinion this is not some occult danger, I’m not necessarily right about that. And if people adopted a custom to be stringent about it, perhaps there is room for such a custom.

As for Arukh HaShulchan, exactly. You see that he changes the Talmudic law in light of observations of what happens to people or animals in his environment. See a lecture by Rabbi Ovadia: http://www.ykr.org.il/modules/Videos/page/3299 See the Shulchan Arukh and Tur, Yoreh De'ah section 116, which deals entirely with things forbidden because of danger. And see Arukh HaShulchan there, for example in subsections 10 and 11 and elsewhere. All the halakhic decisors speak about changing circumstances (see, for example, his reference to Magen Avraham, Orach Chaim 273:1, and many others). By the way, regarding uncovered water, the Tur, Shulchan Arukh, and Arukh HaShulchan write that this is because snakes are not common. In the quotation you brought, he says something more radical: that we can see among us that it does not cause harm.

Nowadays there are many things about which there is concern whether they are harmful to health or not—for example, consuming food products with preservatives or food coloring or artificial sweeteners, etc. But even if there is some health damage in these things, the average person does not avoid them because of health concerns, unlike drinking kerosene or bleach. So wouldn’t it be correct to say that the prohibition of foods because of “you shall diligently guard your lives” applies only to things that the average person would avoid because of health concerns? Then we could say that someone who is extremely concerned for his health may also avoid eating fish together with meat, just as he can avoid preservatives and artificial sweeteners, but not that we should say there is a halakhic / moral obligation in the matter.

The difference is between things that the Sages explicitly prohibited, where there is a prohibition, and dangerous things that were not explicitly prohibited. In things that were explicitly prohibited, there is no room to weigh how dangerous it is. Only if we reach the conclusion that there is no danger here at all and that this was a mistake, can we set them aside. But with things that were not explicitly prohibited and that we assess as dangerous, there is room to discuss how much risk is needed in order to prohibit them. The rule among the halakhic decisors is “The Lord protects the simple,” meaning that whatever people in general are not careful about, Jewish law also does not obligate one to guard against. I don’t think eating food coloring, etc., is more dangerous than driving a car.

Regarding considerations of evil spirit, see also:

http://onegshabbat.blogspot.co.il/2016/01/blog-post_29.html

Continuing this topic: regarding washing the hands with water between eating fish and meat, is that obligatory?

No.

And what about mayim acharonim, is it indeed obligatory?

The halakhic decisors are divided on this, and even in the Shulchan Arukh he brings both views, although anonymously he writes that there is an obligation. One who is lenient has what to rely on, especially since the reason for mayim acharonim probably does not apply today (Sodomite salt).

So with anything the Sages based on a mistake, are we not bound by it?

From what I wrote, you could see that in my opinion, no. Only in a clear factual error.