First of all I think you’re an amazing sister and that he could really use your help but he’s just too scared to ask. Honestly I think forcing him to apply to unis or to jobs might trigger his anxiety and I think that’s not the way to go. First get him to get his confidence up. Starting at something small like just talking to him about how he feels about his stutter and the impact it’s had on his life and start slowly working your way up. Rock climbing sounds like a good and fun idea! Maybe in a few weeks he can order some food at a restaurant or ask a store clerk if they have a shoe in his size. Slowly but surely build his confidence up until he feels that he’s ready to apply to jobs or to unis. Hope this helps.