Demolition derby. Hear me out here, I really think this covers plenty of bases.
The candidates are required to construct their vehicle in a set amount of time before election day (no early starts). They may build it entirely from scratch or accept a standard chassis from the government (if the standard chassis is taken it must be used, no selling it to fund a better all custom vehicle).
The candidates are permitted to assemble a team of engineers to build the car but they must drive it themselves. Other than the free chassis all funds must come from donations (the candidate, their relatives and any companies owned by them cannot donate).
This tests a number of important attributes for a world leader:
Popularity - a more popular candidate will attract more donations (true a right wing candidate will attract more financial contributions from large companies but a left wing candidate should attract more smaller contributions and engineering students willing to donate their time)
Making quick and correct decisions - important in racing and high-tension situations
Sheer balls - To win they'll almost certainly need to take a risk to force an opponent of the track at some point and do you really want a President who will chicken out at the last moment? (unless needed, see above point)