Your third paragraph raised a good question which I will address first. One disabled person with two healthy people is not a problem. It is the scammers who bring in large groups of 4 or more. Most attractions can easily absorb a group of 3 or 4. It is when that number goes higher that there is an issue.
I have posted before to ignore the comments, and stares, if you are legitimate then you know you are and have no moral issue with using a system put into place for that situation. You should have reported the cast member's comments to supervision. We had a family member with close to the same situation as your mom. Can move into and out of attractions and food places, but long distances and long periods of standing were out of the question. He was accomodated easily as were two of us. Long ago I ditched the judgement on sight alone because my training I received helped do that.
To get a GAC proof should be required. Those that need it will have no issues getting the proof the same as you do for a handicapped tag or placard. Make the GAC with a photo, and RFID, application by mail or in person. This will allow no transfers, unable to be forged and convenient as you can already have it before you leave your home to travel.
Nowadays people just want everything now with little regard for others. And while there are many that do not scam the system, there are those that will.
The "requiring proof" thing is pretty much the root of the entire issue. I'm not going to cite law, because it's not my specialty, but it's been well noted that Disney personnel cannot require guests to prove they're disabled, or to ask them to identify their specific disability. Disney (whether because they're a private entity, or because they're not a "medical professional") does not possess the right to collect this information.
While you are very open with your specific disability, most people who have a serious (yet invisible) illness, ailment or other disability, are pretty self conscious about it. And they definitely don't want to be spouting it off to some minimum wage CM in the middle of City Hall. Heck, people who have things like cancer, or other life-threatening illnesses, barely want to discuss it with their doctors, or outside their immediately household.
I would be in that crowd, should I ever become inflicted with such an illness or disability. I'm very private, and I'm not about to tell my deepest secrets to some random Disney employee. As would my dad, who is quite literally in such a situation currently. His solution is to simply admit that there are things he can't do anymore - rather than ask for accommodation - but that's just his personality.
There are loopholes in every set of rules, and that's why I wish Disney would think outside the box. I've suggested (on here) staffing Guest Relations with a medical professional, perhaps in a secluded room, where guests can apply for a specific GAC while communicating with someone carrying credentials and a vow of confidentiality. It may not be possible, or get around the laws, but it would solve a lot of the abuse issues.
Not only would they be able to continue their excellent record of accommodating guests who truly need assistance, but they'd be able to do so in a more thorough and private manner....while also deterring the abusers from even attempting it because of the hassle, or risk of being caught (like the bag check process).
There's no argument that the system is being abused. And I honestly don't believe there's anyone participating in this discussion who thinks the abuse (whether by this group of "elite" individuals, or the general public) is acceptable. It's just that laws and other oversight have opened up glaring loopholes in the system...and I, along with many, feel that Disney needs to find the counter-loopholes.
With any luck, the negative press from this story will actually get Disney to do something.