Being the belayer means you don't have much control and are mostly just watching closely and hoping your buddy doesn't get hurt. Being in a scary lead situation forces you to concentrate which pushes a lot of the fear emotions aside. Worry and fear take a lot of energy, energy which often can't be spared in an emergency where you have some control over the situation.
The key reason for why ATC don't lose their cool has nothing to do with who is more stressed or has more "skin in the game" and everything to do with the fact that if ATC lose their cool then the pilot is more likely to die... and possibly other planes and their passengers and crew too. In order to save lives ATC have to keep their cool even when everyone else around them panics. That's the real crux of the matter.
ATC can have a larger stake than any single pilot. Consider a mid air collision scenario, for instance.
They will handle more lives daily than a pilot. Don't dismiss it.
ATC involved in incidents can suffer from severe PTSD.
One example is WWII commanders would sometimes continue to bring ammunition to machine gun nests even when just adding it to large piles. It may have been the best option when they start, but they never revisit that choice. The same thing happens to pilots where people investigate a crash and find pilots keep doing the exact same thing that’s not working for several minutes without considering that perhaps some assumption they have is wrong.
Finally, some people don’t seem to react much at all to such personal risks.