I love my GameCube. I would be really peeved if this 20 minutes thing happened to me.
The first thing to try is to pull out any memory cards, only use one controller and most importantly: use a known good game disc.
I have a brand new GameCube Game which I only use for trouble shooting. I don't like the game and never, ever play it, unless I need to test something.
GameCube can be finicky if your favorite disc has any scratches, fingerprints.
Discs can also go bad if you write your name on the label side with a permanent marker.
Even if you have taken great care of your GameCube, you may have left it plugged in to the wall outlet (power mains). I know this sounds silly, but the official instructions tell you to unplug the unit when not in use. Sure it's crazy, but I actually do this.
And it is not enough to use a surge protector, you have to make sure your surge protector is still working and that nothing else is plugged into it.
My town has lots of brown outs, where the lights dim and come back on bright. No surge protector will save your GameCube in a case like that. Repeated electrical surges and brownouts can slowly fry any gadget.
If the fan is working okay, you can try this last ditch test. Get a rubber ice bag for headaches or sprains, or a blue ice cooling pack for lunch bags and use it to cool the GameCube while in use. Be careful not to block any vents and watch out for moisture or water.
If the ice pack extends the playing time beyond 20 minutes, you may have to replace the thermal paste on the CPU heat sink, the fan, or make other repairs.
Luckily GameCubes are also available refurbished for about $30 USD if you can not make the repair. You can also find them on freecycle.org if that group is in your area. Best wishes on getting that GameCube fully operational.