It probably can't be fixed without modifying the code dealing with menus and transitions between game and menu windows. The menus are contained in separate GDI rendered windows (each submenu actually shows new window above the previous one) with some DirectDraw bits slapped in between.
You're not the only one noticing this behavior on Windows 10. The menu window is actually active when you press ESC, just obscured. Interestingly, it still works as intended on Windows 8.1.
I noticed with standalone Blue-Shift (could be the same for Half-Life 22.214.171.124, only have 126.96.36.199 for reference ATM), transitions magically start working properly if you change accent color in Settings app->Personalization->Color, but only if the checkbox Title bars and window borders is checked and you have already loaded the saved game or started the new one, just so the graphics stuff is initialized.
This is all in OpenGL mode. Can't see the menus in Direct3D after pause, unless I Win + Shift + Right arrow the window to the 2nd screen.
Blue-Shift also changes cursor shape a bit and for some reason, mouse movement while in main menu is really choppy. I don't remember this on older Windows 10 builds, though I don't exclude the possibility that there's something odd about my particular installation (or that it did happen, but forgot about it...).
Color depth setting shouldn't make difference on modern Windows in terms of transitions. From what I can tell, Windows exposes 16-bit color depth the fist time application or game is launched and pretends it's available on subsequent launches as well if app tried such color depth setting, otherwise it's unavailable on subsequent launches and API calls made on behalf of the app to change screen color depth to unsupported values fail.
If you use that patch, adding -32bpp parameter to the game's shortcut target field will also affect menu screens, which were hardcoded to always switch the screen to 16-bit color depth. Either way, since Windows 8, everything is always handled in 32-bit mode.
Half-Life's registry settings for menu resolution and color depth indeed don't do anything.
The observations I described were made on system running Windows 10 Build 18363.476. I mostly use Win key to make the menu appear normally.