I work at a performing arts institution, and we're continually assessing the entry/audition criteria. We're very concerned about diversity, but a big problem for us is that the state education system's arts instruction does not prepare students to study the subject at university: basically, students have to have private lessons and/or attend an arts school to get to the required level. We can broaden our idea of the sorts of skills that are needed for entry only so far: eventually, we wouldn't be able to get students to the required level in the time we have available.
The institution is elitist in the sense that it accepts only the best, but it already offers free courses to help prospective students prepare for the entrance exams, and there are other initiatives in place to help students access performing arts schools. But beyond the obvious problem of financial affordability for families, it's a bigger question of valuing the arts enough, both as a society to invest in them (say, by raising the standards in schools, or not dismantling school-connected lesson programmes), and as families, to take advantage of the programmes that do exist, or to seek out opportunities. Not surprisingly, many families don't prioritise this...I think that addressing poverty as a society would have a far more sustainable effect than diversity initiatives of institutions, and I think it's not really fair to blame post-secondary institutions for a much bigger problem that starts far earlier in children's lives.
Another, related aspect is that the required skills or even the field itself has to be on prospective students' radar, in the sense that they need to be aware that it's even possible to study the subject in question. There's apparently a world shortage of orchestral bassoonists (unlike e.g. violinists, where there are hundreds competing for every orchestral chair), but how do you find out what a bassoon is, as a musically interested child whose parents don't know what a bassoon is, either?