Finally I found good solution for exact problem. It seems that EVE Online’s new Launcher is written with Electron and I found similar problem on the Net where people are failing to run Electron apps. The offered solution is to run Electron apps not with wine executable but with wineconsole.
So working solution is to replace wine with wineconsole in shortcuts used to run EVE Online, that’s all! In my case changes look like this:
changed to this:
