All these stories have in common that Disney is indeed putting liability above guest experience. In fact, the whole "look over the menu" is putting the onus on the guest, instead of sharing it with the restaurant staff.
OP, I hope you do contact Disney and let them know the manager was blocking the door because she wanted you to sign papers *before* getting medical attention. I also hope you let them know how ridiculous it is for Disney to force you into the "allergy" dessert when the regular dessert didn't have any fish.
Disney talks a lot about industry standards. I don't have an allergy (more like intolerances), but I assume (correct me if I'm wrong) that regular restaurants don't just hand you an allergy menu information without having a chef talk to you. And restaurants don't force you into the gluten free option because you have a fish allergy. What are your experiences at home? I think that should be mentioned to them too.
Experiences at home run the gamut...the best experience I ever had was Chick Fil A. Their registers (at least when I asked on a trip in TN) are set to print out the ingredients to each of their menu items (literally every one), and the person who took my order told me she would special label it and refund me if when I looked at the ingredients I found one I couldn't have. Now, I can't eat very much there, but I am very happy they took my needs seriously...and we have gone back to our own local chain and gotten similar experiences.
With a severe dairy intolerance (to all parts of dairy), I eat at a lot of Asian places when we got out (the most options), and the waiter/waitress will always check for me if something adds something that isn't in the "wikipedia-equivalent main recipe" for a dish. They never mind, and I haven't been sick this summer eating out.
At hotel free breakfasts is where it's been a big struggle - Residence Inn specifically told me that something had no milk (their oatmeal)...and after 2 days of eating it, I went from feeling subpar to violently sick...I left the hotel, went to TN, and instantly cleared up the next morning when I stuck to my own toast at breakfast. I had talked to the front desk and the people in the back...but I'm sure someone was contaminating it with a spoon that touched milk...or they were swirling something they thought was non-dairy but had dairy components (like non-dairy creamer or something, which does have dairy)...so, now, I skip hotel coffee (since I don't trust that their soy milk hasn't been similarly contaminated...and I don't drink black), I skip anything really but fruit and prepackaged PB (I had tea and fruit salad every morning when I was taking care of my mom at a hospital this summer, b/c I couldn't be sick, then)...
And I want to make one note about allergies and intolerances. Allergies are life effecting (aka, right at the instance) and intolerances are health effecting (aka - they will destroy your health in the longer term if ignored) - in the rapid build up of my intolerance this spring, I moved into "serious health issues" category all stemming from the as-yet undiagnosed severe intolerance. So, for many people, they are a huge deal. Ignoring them or downplaying them are the equivalent of telling someone it's okay to smoke cigarettes every day b/c it won't kill them that day.
PS - Colonial Williamsburg just lost a federal case for ignoring someone's severe food intolerance...so changes have happened in that court district regarding treating them with the seriousness they require (aka - now letting folks bring in their own food, etc)...Disney already lets you bring in food, so they were ahead of the game on this issue.
PPS - I also have to give a shout out to one cashier on McDonalds. We were traveling to my mom's and a MD cashier took my needs so seriously, she brought out my big mac no cheese herself, then came back 2 minutes later and made sure it was right. She was a gem, but that's not the "norm" from McD's
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