KAUST vision is to be on top 5 schools around the world in the coming 5 years
also you have mentioned a concern
" Should I do a PhD in a not widely known non-western institution, but in a great group with great funding? Should I be concerned about my career prospects in western countries, or is the institution's reputation nearly irrelevant compared to my research output? Should I rather aim to find a PhD in a well-known top place, if that is what matters for me?"
let me echo this for you as you are heading toward academia
it's not about what institution you graduated from
it's about what you publish and provide to the scientific community
if i were you , i would choose KAUST with a blind eye . they will take a good care of you
and you will end up making money also !
besides . the region is heavily growing into the new silicon valley and technology hub of the MENA region