[+] -> "Scan folder for GOG games"

With this I am assuming you're using Galaxy. First thing; use Windows Explorer to double check that the files are there and that the external drive has the same drive letter as before, and check the specified path in Galaxy if it's the same.

Windows will usually alternate between the available drive letters automatically and the external might not get the same drive letter as last time, but you can force Windows to use the same drive letter on an external harddrive. Like E:\ or W:\. This should be persistent, meaning with a reboot it will remember it. My advice is to use A:\ or B:\ for external drives you use very often (or always) as these will not be used automatically by Windows, only those after C:\ -->.

A and B are remnants from an old time.

I would highly advise against installing games to removable media. First of there will be big speed issues, even usb c is likely far slower than good internal drives. Second, games rely on registry entries, third party components, and their own folders. Unless a game can be made portable you could have all kinds of issues with it. Just copy the offline installers to the hard drive, and install it on the laptop. You can keep saves on the external media if you want to transfer across.

Some people have laptops with very small non-replaceable hard drives. Sure, far from ideal; but sometimes it is what it is.

Not really the type of machine to be playing games one then?
I understand there are always limitations, but it’s like bashing a nail in with a ballon, don’t complain if it blows up in your face. Get appropriate hardware.

First, USB C is only a connector/interface and has nothing to do with speed. Second; the USB 3.x specification (the standard for both connector and speed, as you probably meant) is far from slow and perfectly usable for big games (at least those GOG has). Even USB 2 works just fine on smaller games, especially DOS games. And third; as long as the UNC path is set correctly, there shouldn't be any problem there either. But yes, if one doesn't manually set the drive letter before installing anything, it can lead to some grey hairs later on :D

EDIT: Unless my memory is wonky, SATA 3 has an upper limit of 6 GB/s transfer speed while USB 3 has 5 GB/s (theoretically of course), so why even worry about speed there?

Yes, I meant the usb 3.x spec, but if it’s a small capacity laptop it’s unlikely to have fast usb connectivity. He didn’t mention the game in question, if it is a small dos game, then surely space on the machine wouldn’t be an issue? If space is an issue, why bother with galaxy?

Anyway, some older games can be probabilised and may function ok from usb media,. I still wouldn’t though. I had Diablo portable for a while on 3.0 usb, worked ok but nocd stopped working after a while.

But yes, my advice most defianately is install it locally, outside windows folders.

He has written nothing about his laptop, besides USB 3 has been standard on laptops for years now.

I've done that "portable" thing to the death in the past on FireWire, USB, IDE, SATA and ESATA on OS's, games, and other programs... and trust me, it's a fun ride. Loading/saving times might be a little longer (depends on the game if it require much reading/writing from/to the external drive, some times faster with write cache on), and it's actually more important to choose a fast disk; SSD > 7600RPM > 5400RPM.

A good example is Witcher 3 that I have on an external Toshiba 3TB drive; it only takes a few more seconds to load in, the rest is up to the mamaboard, memory, CPU and GPU.

In any case it's his choice. Some like it, and some don't... and that's perfectly ok :D