Looking on Google play store for apps that would add a desktop/home cast icon/shortcut, I have found the Cast Menu Widget.
And inIn order to get the quick settings button, the (under 1 euro) paid version of the above tool can be used - or, as said here, one can install Nugat/Oreo Quick Settings.
Both buttons (1st click) will first show the list of available devices (2nd click to select one), then the last confirmation option, "Start now" (3rd and last click).