Does that exist?
GOG is a store. What you're wanting is some frontend app to do downloads and act as a launcher.
GOG has some frontend, GOG Galaxy, but GOG doesn't do a Linux release of it.
I personally dislike using Steam's client as launcher, and don't like that model (I've got better tools to launch programs and would just as soon not have Steam interjecting itself and would rather not even having it run). One of GOG's selling points is that once you buy a game and download the installer, you don't have any dependency on GOG
if the company goes under, you still can play the games.
For GoG, there's a command-line program, lgogdownloader, that can, among other things, batch-download all your games, but I don't believe that it will auto-install them (and in fact, I don't know if all of the games on GoG have installers that can do headless installation). The open-source, command-line downloader and not having to run software from the company is more-or-less my ideal model, though I can understand wanting to do a headless install.
EDIT: It does look like GOG Galaxy can run under WINE, which is how you'd likely run most of the games you're obtaining from GOG, so you could just use the Windows client in WINE. I suppose that you'd download the Windows client here if you wanted to do that.
EDIT2: The Wine AppDB entry says that you will need corefonts (a collection of Microsoft fonts) to run GOG Galaxy, so if you've never used WINE, setting up a 64-bit WINE environment in ~/.wine and installing corefonts in it will look something like this (on a Debian-family system):
$ sudo apt install wine winetricks
$ winetricks corefonts
Download the GOG Galaxy binary for Windows from its website.
And then run it:
$ wine GOG_Galaxy_2.0.exe