I really like how all of gog.com's games come with a uniform style icon for the provided shortcut, and I was really hoping there would be a way to make my own shortcut icons for games that aren't available on gog.com in that style

I assume you mean the ICO file with an image inside a white ring.

I would use a free program called GIMP, but there are others, Windows paint won't cut it. Main thing is you want a program that gives you control over three things:
1. Transparency (alpha channel) - because the image is shaped like a circle with the background (your desktop) showing around the corners of a normally square icon.
2. Circular selections - to select pixels in the image in the shape of a circle so that you can mask off the center you want to keep (invert selection) and erase (make transparent) the pixels outside of the circle.
3. Layers - so that you can overlay one layer containing the white ring on top of another layer which contains the image that you want inside the ring.

Make a ring image first and keep it separate so that you can reuse it for all future icons.

The layer containing the ring will have transparent pixels outside and inside the ring. The layer containing the image of your game will have transparent pixels around the corners making a circle. So when you put the ring layer on top of the the game image layer you'll end up with your desired icon. After you put one layer on top of another, you can merge the layers into a single layer.

Then you export to Windows Icon ICO. An ICO file is an archive containing one or more image files of different sizes, ie. 16x16, 32x32, 64x64, etc. If you open a GOG icon in GIMP you will see all these sizes as separate layers. You don't have to make all these different sizes however, just the one(s) that will look good on your screen as the OS will scale the available sizes if needed.

It should be possible to automate this process with a script. I am not a graphic artist, so someone else might give you better instructions.

I wouldn't bother with GIMP unless they've majorly uptuned their UI.

Instead, I'd suggest working in vectors, as any modern icon would be made in, using Inkscape. That way, the icon can scale to whatever size you dang gosh well please.

Or if you would insist on a raster program, use one that isn't being developed by obstinate coders whose stiffness towards their project has been preventing a major overhaul release, like Krita, which is made by the KDE Foundation. Much like DosBox Staging, I actually see a future in Krita, unlike the GNU Image Manipulation Program.

It was late at night, so in my instructions, I forgot to mention that rather than designing an icon from scratch, the starting point for this procedure would be some image grabbed from google image search or pcgamingwiki box art - therefore I recommended a raster program. A portion of that image would then be cropped to a square and scaled down to the largest desired icon size such as 128 or 256 pixels. Since the OS would do scaling down to 64, 32, 16 pixels if desired, I did not consider scaling a priority. Like you said there may be more user friendly programs than GIMP, without watching tutorials someone using it for the first time will become frustrated as it will seemingly not respond to commands.

One other thing I failed to mention is that rather than preparing a white ring as a separate image to overlay over the image grabbed from the internet, it should be possible to make a white ring on the fly with a stroke from path or stroke from selection (circular). That could avoid some extra steps. The only thing is that the gog icon ring is not a simple white line, but it has shadowed metalic edges to make than stand out against a white background. How to make these edges while stroking a path, I am not sure without some investigation.

Rather than create the icon 100% from scratch, you can cannibalize an actual gog.com style logo. Just install any of your GOG games, then go into the game's folder and look for the file "webcache.zip" In all cases, the PNG files are the gog.com logo for the game in question.

If you open the largest PNG image in any image editor, you can just erase the existing art then replace it with the art for the game you need an icon for.

Additionally if you have access to Adobe PhotoShop, it looks like the Blending Option "Inner Shadow" was used by whoever makes the actual gog.com icons.

I have a memory that like every game back in the day came with an ico file in it's install that would have been a good starting point for editing, but checking some of my non-GOG games that doesn't appear to be the case, though some still do.

How about crayons and an eraser?