Unfortunately I don’t know that going to a medical model would change much. People willing to lie or exaggerate to a CM would presumably be willing to lie or exaggerate to a doctor. The same social media groups will pop up with advice on what to say and what diagnoses are most subjective and easiest to claim. And those abusing the system may feel more emboldened, because they will have something official in hand, vs. the current system where it’s at the discretion of the CM. I think the key issue at the moment is that DAS overlaps so much with a paid product (Genie+). As long as that financial incentive is there, there will be a small but persistent group of people determined to cheat the system (I actually don’t think it’s a substantial number of people, though.)
That's the rub. It's quite easy to get a note from your doctor for any number of conditions, even if you don't have them or they aren't impactful enough to qualify for accommodations. If Disney were to implement a program identical to Universal's, it would create additional barriers to those with legitimate disabilities, while doing little to curb those without them from getting into the program. Some guests with legitimate disabilities would go on vacation without documentation and may potentially be denied the accommodation as a result, while guests who know how to game the system would have done the work in advance to procure a note.
Thankfully, from what I've heard, Disney's plans include contingencies for guests who do not have a formal supporting documentation with them, and should in theory be able to accommodate guests who apply in-person without any pre-preparation.
Abuse currently occurs in sufficient numbers that it creates barriers to selling more Genie+ and creates guest dissatisfaction amongst Genie+ purchasers due to increased LL wait times. Even though the LL wait times are not that much longer than when they were FP+, now that it is a paid program, guests expect very little to no wait. A 15 minute wait went from being acceptable when it was free, to unacceptable now that they're paying as high as nearly $40/person/day extra for the privilege.
The real problem, as you correctly point out, is that there is such overlap between the premium product and the accommodations available to guests with certain disabilities. So long as they are substantively the same, and one is free while the other is paid (there's no way they could make DAS paid), this will continue to be a problem, though I suspect that requiring evidence of disability will deter some of the more lazy abusers.