I have a "USB Transfer Cable" that my brother and I used back in ~2006 for transferring files to a project PC we built. It was advertised as being red but when I opened the box I was surprised to find a thing that looked like a hot-pink jump rope. Crazy looking contraption, but it worked great. Much less fiddly than networking, and was much faster for large transfers than using the flash drives available at the time.
I just dug it out of my box of USB cables and on my main Windows 10 machine it is still detected and installs itself as a CD-ROM drive called EasyCopy-Net, or in Device Manager it is listed under CDROMs as OTi EasyCopy-Net USB Device. When you "open" the drive it always runs the application for transferring files. It actually works incredibly well.
Crazy thing doesn't have a name brand or any markings on it anywhere, so I don't know what it was actually called (or where I bought it) but, it does seem to work, and I vaguely recall it being labeled as a Windows XP device. I think it requires USB storage drivers to be installed on older operating systems. I just connected it between my Windows 10 system and my Windows 98SE tester system (which has mass storage drivers already) and on the 98 machine it was detected as a USB mass storage device. Once the mass storage driver finished installing (I just had to click next a couple times) it then found the EasyCopy-Net device which proceeded to install itself as a CDROM drive with no interaction. When I open the drive, it runs the application and I can see all the files on my Windows 10 machine. It's kind of crazy how it lets this old 98 machine have completely free access to all of the files on all of my drives. Even system files and user data folders are totally accessible (though I'm logged into the only password protected account, so I'm not sure how it handles accounts that aren't logged in currently).
Anyway, yes these devices exist and they work. Now that I'm reminded of how convenient they are, it makes my cobbled FTP sharing setup here seem a bit redundant, since I'm only ever sharing files from about 3 feet away, which this thing does without any fuss at all... 🤣
... as to whether you'd be able to make this work with no interaction with one of the PCs, I highly doubt it. If this is for some kind of kiosk at a retail store or something, there has to be some other way to interface with the machine. If there isn't, then I don't know how it was ever even installed in the first place. The most astonishing thing about this whole scenario, is that a system that cannot be interacted with in any way (no display or inputs) has been operating since Windows ME would have been relevant... and apparently no one has had to ever service it??? An ME machine would presumably only have USB 1.1 support anyway, so even if you could use a hub to connect a USB display and a keyboard, I can't imagine it'd be too usable on a 12mbps shared interface.
If the system can have a monitor and kb\m connected temporarily, the scenario gets more feasible, but I'm not sure if any of these transfer cables are quite as transparent to the host PCs as you'd like. For example, when I access mine (G: drive) via the command prompt or by going to that drive in Explorer, it just shows it as a CD with an autorun.inf and an .exe file, occupying ~400KB with no free space. So, it doesn't seem to work in the way you'd need it to. If you can tell the machine to put it's files into a specific folder on the machine's hard drive (rather than to an external drive), then a transfer cable like this would at least give another PC access to those files remotely through the USB port. They won't be dropped directly onto the remote PC however. Also, the application does need to be running on both computers, but as long as autorun is working simply plugging it into the headless system would run the application and allow transfers to and from it.
Time Machine = FIC PA-2013 2.1 - K6-2 500 - 256MB PC-100 - TNT2 Pro 16MB AGP - Labway Yamaha YMF719-E - Midiman MM401