It’s true that there are some apps not directly associated with any DE that have moved to GTK4. If GTK5 actually was likely to come out any time soon you might have to worry about finding alternatives when they switch to GTK5. That being said though, GTK3 had been out for over 9 years before GTK4 came along, there were 4 years between the “last” version of GTK3 and GTK 4.0.0, they’re still in the “oh, this is what we’ll probably do when we release GTK5” phase of development, XFCE has already made a bunch of progress porting everything over to Wayland, and DE agnostic apps are less likely to switch over right away if mid-size DEs like XFCE and Cinnamon still don’t have good Wayland support. I wouldn’t stress out about it too much.
If you have the default version of Mint installed then your desktop environment is Cinnamon. There are also XFCE and MATE versions, but you have to go out of your way to get those. The default file explorer for Cinnamon is called Nemo, so if you haven’t changed it that would be what you are using.
Honestly, I think your best bet is trying Disks or maybe gparted if you like cli apps, and setting a mount point for the device from one of those. Linux doesn’t always like NTFS, but you should at least be able to mount and read the drive consistently, although I have to admit I’ve never used an NTFS formatted external drive, so maybe something weird is going on with that.