Issues, Workarounds & Localization

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Sorry for the late reply, kind of lost track of this thread.

So, after looking at the log file you provided then it looks very much like the problem of some 3rd party software interfering with the HTTPS connection the launcher wants to establish. I've just observed the same behaviour here when I run the EVE launcher with Fiddler (a network debugging tool), because Fiddler cannot handle the request to the payload.

You said previously that you tried disabling all your anti-virus and similar software, but the symptom you describe in combination with the log file just indicates that there is still something that interferes with the connection. And just to be sure: You did disable these programs before starting the EVE launcher? Disabling them once the EVE launcher is running is usually too late.
There is also a great list of software that is known to cause issues on the steam knowledge base at https://support.steampowered.com/kb_article.php?ref=9828-SFLZ-9289 - basically most of the things suggested there count for the launcher, too.

Except for comparing what you have installed with the list on the Steam knowledge page? Not really. There are countless "utilities" out there which may cause trouble and some of them only cause problems in specific versions with certain settings enabled or disabled.
It could also be your system proxy settings but I am sure you already checked these.

And this is the only thing I can tell you: Something is tampering with the HTTP request the launcher sends out to get some information from our content delivery network, e.g. the launcher does not get a useful response. If you are lucky the request never actually went through and the connection times out, which will make the launcher try to fall back to a precomputed update, but there is no guarantee that such an update exists from your current EVE client version to the latest version.

Do I understand you correctly that you had to use that software in order to get the launcher working? Or did you have to disable it?

That's weird! Only thing I can imagine that starting it overrides whatever broken proxy settings there might be lingering around and thus makes it work. But good that it solves your problem, even though such an extra utility should not be required to run the launcher.