Maybe, but an actual ADA disability carries a lot more weight than wanting to feel more relaxed. The law requires that “reasonable accommodations” be made for disabilities, and running shoes are more than reasonable accommodations for a veteran missing a leg. Management probably would have offered to amputate the leg from any intern who wanted to be allowed to wear casual shoes. Then they would have pointed out the clause in the interns’ applications where they acknowledged that they did not have a disability requiring that kind of accommodation, and shown them all the door anyway.
That's true in the real world, where there are laws and stuff, but in the bubble of school, there is no acknowledgement of reality, only "feelings."
At their schools, these students are treated as equals (at least!) to the faculty/staff, both because of ideology and because the systems need them and their parents as customers. At a school, they could expect a sympathetic response to the "merits" of their dress-code complaint, as well as praise for their tiny cleverness in presenting it neatly in writing and their tiny communism in acting as a group to exert pressure.
However, they've never actually had to mature. Their dress-code issues are simply, "Mommy won't let me wear what I want all the time. WAAAAAAH!" dressed up. If it hadn't been dress code, they'd probably have found some other issue that "required" them to challenge what they perceived as illegitimate patriarchal authority structures.
The work world (outside government) was obviously a rude awakening for them. I wonder what percentage of the group will ultimate choose to get real and participate in life as responsible adults, and how many will try to organize their lives so that they never again have to leave their "safe space."