In short:
No it doesn't work on WebView2.
Why?
Because WebView2's rendering architecture is problematic for Steam's overlay to hook into.
So Scirra came up with a workaround to make Steam's overlay work. This is described in the Steamworks plugin documentation.
It adds the D3D11Overlay_x64.ext.dll to your exports as you've noticed, which is a library used to create this DirectComposition Direct3D 11 overlay-frame, on top of your game. Steam then hooks into that, instead of your game.
While this makes the Steam overlay work on Webview2, it also breaks direct game recording with any recording tool, including Steam Recordings, as well as Steam Remote Play and Steam Broadcast. As applications now pickup this transparent overlay, instead of the actual game.
Scirra likely can't do more than this -> Steam doesn't care because not enough people report it to them -> There will realistically speaking never be a solution to this.
There's a small chance that either WebView2 or Steam gets an update adressing this in the future, but i definitely wouldn't count on it.
A workaround would be to use 3rd party exporting solutions like Pipelab, which uses Electron. Use an old C3 version to export with NWjs. Or migrate your game to a different game engine.
Which is unfortunate, but it's the situation we're in.