No, it’s not “better than nothing.” In fact, it’s worse than nothing. It’s providing people with misinformation that could LITERALLY kill them. If the information on the menu can’t be counted on to be reliable, it should not be handed out to patrons.
OP, I’m so sorry this happened to you. My niece has the same dairy allergy (in addition to sesame, tree nuts, and unbaked egg, and her sister is allergic to wheat and peanut). I know everything we go through to make family gatherings safe for these little girls...I can’t imagine what people like you and my sister-in-law go through on a daily basis to give these kids normal lives while also keeping them safe!