Providing salary to interns would encourage less affluent applicants, but it would also make the positions more attractive to everyone, increasing competition.

You have this exactly backwards. The social trend being protested is "removing salary from interns would discourage less affluent applicants, but it would also make the positions more clearly intended for the social elite." Give the phrase "rise in unpaid internships" a google. Then do an image search on "unpaid internship graph." The simple fact is that paid internships are extraordinarily rare these days whereas previously they were common. It's also a fact that the positions being held by the unpaid were formerly held by the paid. Within my own industry, I've worked with unpaid editors, unpaid script supervisors, unpaid dialog editors... I've never worked with an unpaid PA, though, 'cuz nobody does your scutwork for free.

Kathleen Kennedy is worth looking up. Pay careful attention to this passage:

During the production of 1941 (1979), while working for screenwriter John Milius, Kennedy came to the attention of Steven Spielberg. Spielberg hired Kennedy as his secretary, but both she and he reported that she was a terrible typist who was kept on only because of her good production ideas.

Not mentioned in there is the fact that she spent several months hanging out in front of the office, begging for work every day. Nor the fact that eventually they let her in to make coffee. Nor the fact that she didn't get paid well into the production of Raiders. Not to diminish Kathleen Kennedy's skills in any way, but the fact of the matter is, if she hadn't hung out with no income for months, she never would have become Kathleen Kennedy.

Success is being in the right place at the right time. The way things used to work, you could earn a pittance so that you could spend a long time at the right place. The way things work now, youth are led to believe that hanging out in the right place is reward enough for spending an impossibly long time there. My buddy? Well, for starters, he comes from old money. And his wife worked, so there was that. But three years of 80 hour weeks (for free) nearly destroyed his marriage, resurrected a drug addiction and annihilated his health. Meanwhile, that's six man-years of income out of the labor pool.

It is wrong to not pay interns. When a job that is typically remunerated is offered to someone in exchange for prestige and the promise of advancement, taxes aren't paid. Livings aren't supported. And opportunities for equally (or better) qualified individuals who do not have the luxury of a rich uncle or forbearing parents dissolve in the wake of employee abuse.

This is how gilded ages start.