Non-Disney owned restaurants do treat allergies differently! Their chefs and cast members do not have the same allergy training as Disney owned restaurants and don't have to adhere to the same standards. We haven't had good luck specifically with the non-Disney owned restaurants in World Showcase at EPCOT and many others have reported the same.
San Angel only has a gluten/dairy allergy menu. I'm in a Disney allergy group online and San Angel has horrible reviews, common complaints that staff is not knowledgeable about allergies, ingredients or cross-contamination, chef does not always come out even when requested (which needs to happen where there isn't a top 8 allergy menu) and the allergy toothpick common in Disney owned restaurants is sometimes missing which is a huge red flag. La Hacienda is the Mexican quick service at EPCOT and gets great reviews fyi.
Yak and Yeti does have better reviews, the company they are owned by (Landry's) does an OK job in general with allergies. They do not offer top 8 menus so you won't know what you can have until you get there and speak with someone unfortunately.