I got this resolved on my system (Ubuntu 18.04, Intel i5-3470 iGPU) by installing and using Vulkan rendering. I did not have to download the Sys folder from the Dolphin source repo.
I must note that Vulkan is not supported on any CPU earlier than Ivy Bridge, so this may not be a viable solution for some people. Mine is the earliest possible generation chip this even works on, so I guess I got lucky.
Install Vulkan support and testing application
sudo apt install -y mesa-vulkan-drivers vulkan-utils
Test Vulkan (successful example):
vulkan-smoketest
Modify the GC and Wii cores' config files to use Vulkan renderer:
sed -i '$ i\video_driver = "vulkan"' /opt/retropie/configs/wii/retroarch.cfg
sed -i '$ i\video_driver = "vulkan"' /opt/retropie/configs/gc/retroarch.cfg
After doing this, GC games worked fine. I'll come back and modify this post if necessary whenever I get around to adding Wii to my system.