The fish rots from the head.
In India it's purely score based. Top on the JIT (although I guess they have special things for scheduled castes or something), you get in. In Sweden the same. Top 1% on the högskoleprov and there's nothing you can't get in to. Maybe KTH has its own, more advanced maths test for entry, and I think you need to pass an interview for medicine at KI, but aside from that nothing you can't get into. At ETH they see whether you can pass a first year of courses.
But in the US it's a bunch of weirdness, and it's from Idpol. I don't know if you could avoid it-- India obviously doesn't want to avoid it completely, but another idea is to not have the arbitrary eliteness be a thing. I used to believe in something like it, believing that only certain Swedish universities were okay (and that's probably still true for education), and then a bunch of Germans who wanted to be professors got jobs at the ones I thought were shitty and started doing good research and suddenly you ended up with places like Örebro, which I had regarded as 'what even is this' producing real science. I think the Germans are right and that a distributed less status-based university system is sensible.
Furthermore in the current political environment, such analysis-free rants aren't just chum that makes like-minded rambling uncles need more blood pressure meds, but rather can end up being fuel for someone-must-do-something-type destructionist rallying cries that only serve to facilitate more grift by the performative strongman administration - compounding the very problem!
Constructively, the difficulty is that reforming institutions and restoring societal trust is very hard. Here we've got at least four things that need to be done simultaneously -
1. restoring belief that the system will significantly punish you if you lie/exaggerate about having a disability
2. restoring trust in the system such that people, both internal and external to the institution, aren't inclined to panic over "xx% of students claiming disability"
3. reforming the general system for people without disabilities, eg testing methodologies and cramped housing accommodations
4. generally reforming what counts as a disability that makes sense to even try and mitigate
Fail at doing any one of these and we've still got similar pressure to cheat, so the problem will only ever retreat a bit rather than having formed self-reinforcing cultural values.
(I'm addressing the problem referenced by the article, not the adjacent problem you've described)
I think the right solution is strict test-based admissions, like in Sweden, and forbidding schools from admitting other than on test results. In addition to that, making things like getting top government jobs also test based. In Sweden we do this for diplomats. You have to get a university degree, but whether you get in doesn't depend primarily on prestige or interviews, it depends on whether you pass a somewhat difficult test.
In this way you ensure that what university you went to doesn't really matter, and over time what I imagine is a more distributed university system where co-operation with institutions and someone's particular ideas matters more than what institutions he's associated with, and where the extreme prestige of particular schools disappears.
I think it's especially important to eliminate the hoop jumping, so that people can know that, as long as the extreme prestige universities exist, they can get into them if they perform well enough on exams, with nothing being able to interfere with that-- no individual judgement, no subjectivity, no hoop you haven't jumped. Just pure merit, like the Swedish or French system.