Everything except for the launcher. Even full screen works with my setup. (Maybe try disabling the Steam overlay if you can't get full screen working? I don't know if that has anything to do with it.)
What does not
VC2012 installer hangs, and I suspect this is why the launcher doesn't run. You need to delete the installscript.vdf file from ./drive_c/Program Files/Steam/SteamApps/common/Age2HD/_CommonRedist/vcredist/2012 so that Steam stops trying to install it each time you try to launch the game.
I suspect this is the reason why Launcher.exe also doesn't run, so again as already suggested you need to get rid of the default Launcher.exe and instead rename AoK HD.exe to Launcher.exe.
What was not tested
I just tried a single player random map game (campaigns or multiplayer not tested).
I use fglrx, and have WINEARCH=win32 set in my environment variables
|Operating system||Test date||Wine version||Installs?||Runs?||Used|
|Show||Gentoo Linux||May 02 2020||5.7||Yes||Yes||Yes||Gold||Aristocrates|
|Show||Rosa Fresh 2016.1||May 12 2019||4.8-staging||Yes||Yes||Yes||Gold||Giovanni Mariani|
|Show||macOS 10.13 "High Sierra"||Nov 26 2018||3.0.3||Yes||Yes||Yes||Gold||YVES|
|Show||Gentoo Linux x86_64||Sep 21 2018||3.15-staging||Yes||Yes||No||Platinum||Erol M. Yasar|
|Show||Debian GNU/Linux 9.x "Stretch" x86_64||Jul 15 2018||3.12||Yes||Yes||Yes||Gold||Daniel Bulgakov|
|Bug #||Description||Status||Resolution||Other apps affected|
|29384||Multiple applications expect correct handling of WRITECOPY memory protection (Voobly fails to launch Age of Empires II, MSYS2)||STAGED||View|
|38784||Age of Empires II: wine 1.7.x series do not work with the external monitor as the primary output||UNCONFIRMED||View|
|42435||Age of Empires 2 HD Steam has a dimmed down interface and the ingame main screen is just white with an intel 945gm gpu||UNCONFIRMED||View|
|43758||Age of Empires 2 HD Launcher has missing text||NEW||View|
This is a known problem with older Intel GPUs. Try to use Wine 2.1. In most of case solve Direct3D glitch.
Hit each direction key in succession while holding Ctrl. For instance: