I was wondering if Unix systems use the GPU for the startup splash/loading screen because I've been having some trouble with an overheating Mac with graphics issues. Unix-type systems (such as MacOS 10.6, 10.10 and different versions of Ubuntu) show the splash screen, but never actually boot into the GUI (typically just a plain black/blue/white screen after the startup splash). Windows, however, starts up (I assume this is what's happening as I can hear hard drive activity) and only shows a black screen (no splash or loading screen). This just made me curious as I have a cursed 2008 ATI iMac. I plan later to try reapplying thermal paste to see if that does any good, and then try a reflow (I know this'll only be a very temporary solution but I just want to see if anything will work), but if all else fails, it'll probably go into the bin.
The answer to your literal question is yes: all systems use the GPU to display startup messages and splash screens. That's because going through the GPU is the only way to display something on the monitor.
However, the answer to the question you meant to ask is no: the ways the GPU is used during startup and after the system has fully booted are different. During startup, the operating system uses the GPU in text mode or as a simple framebuffer. These involve very little work from the GPU, so they are unlikely to trigger GPU bugs or to make it overheat. Text mode is limited in that it can only show text in a single monospace font. Framebuffer mode can show arbitrary images, but it's slow. Both modes may use a resolution that's less than the maximum that the GPU and monitor can do.
Once the system has fully booted, it likely starts using the GPU in a different ways, using its computational capabilities. This involves a complex driver in the operating system, and may involve some nontrivial computation on the GPU. Under Linux, this mode is part of the X window system (or a replacement for it such as Wayland.
You may be able to get a GUI on Linux with the X.org
fbdev driver (which uses the GPU as a simple framebuffer) or with the X.org VESA driver (which is a very old standard that does little more than a framebuffer and has limited resolution). It won't be fast, and it might not be pretty, but it's better than nothing.
You may need to work in text mode first to prevent X from starting in a mode that doesn't work. The way to do this depends on the distribution. The Arch Wiki may be useful even if you don't use Arch. Once you've logged in as root, create or edit
/etc/X11/xorg.conf to choose the video driver. For example, for fbdev, you need something like this (untested):
Section "Device" Identifier "fbdev" Driver "fbdev" Option "fbdev" "/dev/fb0" EndSection
You also need to install the appropriate driver (if it isn't already present), which again is distribution-dependent. For example, on Debian/Ubuntu, that's
apt-get install xserver-xorg-video-fbdev