Mosquitoes are the creature that kills the most people on earth. Isn’t that a factor for permitting killing them even on the Sabbath (like snakes and scorpions)?

Regarding what you wrote, “but they permitted it for a need” — what kind of need did you mean? A food-related need? Or a significant need?

And regarding the second point, why does that seem puzzling to you? What is the argument not to permit it based on the reasoning of “since” — since it was permitted for food preparation.

I’m not familiar with those data. If it is dangerous, then of course it is permitted, but it is hard to assume that every mosquito has a presumption of danger. Usually people kill it because it is annoying. In the Talmudic passage I referred you to, Shabbat 3a, it says that trapping a snake and lancing an abscess is exempt and permitted because of suffering (see Tosafot there). It is not clear to me what sort of suffering permits it, though it is brought there that this is obviously not a case of danger to life.
That seems puzzling to me because the slaughtering that was permitted is the slaughtering of kosher animals for the sake of eating, and to derive from that the rule of “since” to killing flies, which is a completely different action, is puzzling. True, there are approaches that mean the entire category of labor — that if taking a life was permitted, then this labor is generally permitted on a Festival of any kind and for any need (not just without reason, but where there is a need) — but in my view that is a puzzling interpretation. There are opinions that the labors permitted are those whose primary nature is for food preparation, and I do not think taking a life is such a labor (though carrying does not seem like that either). One would need to get more deeply into the passage, but offhand this seems like a strange leniency to me.

You wrote: “the slaughtering that was permitted is the slaughtering of kosher animals for the sake of eating.” If I apply to that the rule as Wikipedia defines it: “Performing labors on a Festival was permitted even when they are not for the purpose of food preparation, but rather for another Festival need,” then one has to say that slaughtering too was permitted even not for food preparation. What kind of slaughter is there that is not for food preparation but for another Festival need? That is exactly killing mosquitoes — taking a life not for food preparation but for the sake of preventing suffering. Why do you see this as a completely different action? After all, both are prohibited because of the labor of slaughtering, which is defined as taking a life.

I said that I haven’t gone into this topic right now, and therefore I can’t bring sources or reasons from within it. What I wrote is that on the face of it, this extension seems puzzling to me.
Slaughtering not for food preparation would be killing a chicken in order to play soccer with its head — an inevitable result. But we are still talking about killing a chicken, which is similar to slaughtering for the sake of food preparation. Killing a mosquito is a completely different act (that is not slaughtering, even though, as I noted, in terms of the Sabbath laws it is the same labor — taking a life). This will probably depend on the parameters of the rule of “since,” but one would need to get more deeply into the passage in order to discuss it.