[x] ...their bodyguard
[x] A senior student, assigned to mentor them
[x] Older than he looks
[x] Offer Rin as someone to talk to, if they have a concern or something on their minds
[x] No. The children aren't really his mission - they're just his cover. It's not his place to get too deeply involved
This combination provides Shinji with enough free time and opportunities to improve while being the only member of the team who has to prioritize the mission above all else. At the same time it forces Rin to come out of her unhealthy social stupor and create meaningful bonds with other people. Being seen as older will force people to give more credit to Shinji's words, when the time for inevitable disaster comes.
Hilde may play her games, Rin may be as incompetent as we expect her to be - but Shinji will ensure the success of the mission. After all, that's why we are here.