Everything I've seen for videos of people using vehicles with carbs, the vehicle stalls out on slants and inclines.
This is very true. If you want to run a carb, you need to have a carb designed for off road, or modified. If you run an American V8, they make a "Off Road" carb for them.
Guys that are really into radical off road stuff will convert the engine to Propane, it does not care what side you keep up, it is under pressure and will work no matter what. The engine will starve for oil, but the engine will keep running till it seizes up.
For normal trail runs, mild off road stuff, a carb is fine though. Only rather steep inclines and side hills will give you problems.
Each carb will be different, but once you get past a certain point, the fuel overflows the bowls and you flood out, or it runs the bowl dry and it starves for fuel and dies. Either way it stops running and you have to get back to level ground to get it going again.