They very well might, particularly if English isn't their first language, which is very common. I'm in Texas, so I know how it is - there's a ton of people who don't speak English but have become accustomed to the rules and processes in place and work with those.

For example, they can read road signs. If you put up a new road sign that means the same thing as a previous one, they won't be able to read it. If you replace a "Stop" sign with a "Brake" sign, they won't stop. Because they don't read "Stop" - they know the sign means they should stop.

I'm sure there's thousands of Americans accustomed to reading the "contains:" label who would be blindsided by this. Particularly because today a lot of, maybe most, "dairy" product don't contain milk.