So, I managed to put my hands on two PAL Gamecube games (Rayman 3 and Chibi Robo) and tested them against GeckoOS 126.96.36.199 and concluded that the answer to my original question is yes and no. Allow me to explain.
Rayman 3 worked like a charm. I was able to choose the game's language, play at 60fps and save and load without any problem. Chibi Robo, on the other hand, didn't run under GeckoOS, nor any other method I tried. It may be that my copy is somehow broken, but I can't say that for sure, because I was still able to use it to create a CFG/DIOS MIOS backup (more on this below). If anything, the text on the box of my copy is in German.
The best solution to play import Gamecube games (as well as their backups) is actually Nintendon't, which is an app that you can launch from the Homebrew Channel and doesn't need any kind of additional modding (unlike Priiloader and DIOS MIOS, see below). It also has far better compatibility than DIOS MIOS. Using it, I was able to play my copy of Chibi Robo, which indicates that it has a more complicated type of region lock.
I was unable to use the latest version of Nintendo't on my Wii (downloaded directly from the official repository), however, I found this version, which worked, and through it I updated to the lastest version.
In the end, though, I figured out that the best solution for playing import Gamecube and Wii games involved using Priiloader and/or a USB Loader along with DIOS MIOS. I won't go much further into details about installation, but their basic gists are as follows:
Disclaimer: you are sole responsible for absolutely anything (good or bad) that may happen to your console should you try the suggestions below.
Priiloader is a small application which places itself before the System Menu. Its main feature is to directly boot the Homebrew Channel or a file you can install from a FAT device. Preloader is also able to apply memory patches to the system menu to remove limitations or other things.
Among these "limitations and other things" that can be removed is region locking for Wii and Gamecube games. I believe it might work for 99.9% of all Gamecube and Wii games. In my case I could play Rayman 3 but I was still unable to play my copy of Chibi Robo.
It's a little risky, but unless electrical energy is cut off during the installation, I don't believe there's much of a problem.
- USB Loader
USB Loaders [are applications that] allow you to rip your own discs to an external USB hard drive and then run them from there.
Not much else to say. As the reference says, I suggest using the "Configurable USB Loader" (CFG) application as your USB Loader of choice.
- DIOS MIOS
DIOS MIOS is a modification to your Wii's system software that allows you to play GameCube (GCN) games on your Wii from a FAT32 hard drive
DIOS MIOS is not an application you access, but rather an internal mod that is used by the your USB Loader to run the Gamecube games. In order to install it (as well as all of its dependencies), I'd suggest using this tutorial, mostly because it worked out for me.
Contrary to what the first DIOS MIOS reference says (it may be outdated), newer versions of DIOS MIOS (I tried 2.11) DO allow you to play retail Gamecube games. The problem, though, is that these games are ran under DIOS MIOS and are subject to its compatibility list. This means that if you have a Gamecube game that runs perfectly on an unmodded Wii, installing DIOS MIOS may cause this game to no longer work. For me, it happened to Phantasy Star Online Episodes I and II. As far as I have researched, there is no definitive solution this problem and the best I have managed to get is to install DIOS MIOS whenever I use the USB Loader and uninstall it whenever I play Phantasy Star.
If your System Menu's version is 4.3, once you have installed DIOS MIOS, you can uninstall it by installing this WAD using a WAD manager (e.g. Multi-Mod Manager, WAD Manager 1.7). This WAD is a part of System Menu 4.3 and installing it should replace DIOS MIOS. To reinstall DIOS MIOS, you can simply install its WAD again, it's not necessary to perform all of the stuff in the referenced tutorial again.