Wait, what? Robert was a well-known, likeable choice. Meanwhile Doran was a Dornishman (so as likely to be accepted as a black mayor of Atlanta in 1910), Mace was known as a fool and impressed nobody during the civil war, and Tywin seems to have 2 settings: "rule through fear" and "declare war"
I seriously think that Robert got the job because he was popular and likeable and has a decent claim, and Jon expected to handle intrigue himself. Not a bad plan in theory - although in practice it wildly overestimated Robert's competence as well as his own.