He's not just walking around! He's got crutches. Look, he even uses them.

(He also also considered how well said crutches would go as makeshift weapons. Link's pretty sure he could do some decent damage, if he wanted to.)

As for what point Link was trying to make...he shrugs. He wasn't trying to make any kind of point, really. Mostly he's just...annoyed. Here he is, standing in the middle of a dueling hall...and he doesn't get to compete. He only gets to stand around, watching.