Drop the "every" part and you can see the word that needs to be agreed with.

    "One is supposed to do such and such."
    "Everyone is supposed to do such and such."
    "They are supposed to do such and such."
    "People are supposed to do such and such."

This also applies with "some"

    "Someone is supposed to do such and such."

https://en.wikipedia.org/wiki/Quantifier_(linguistics)

Part of the confusion may be that "everyone" is a single word while the example sentence in the Wikipedia article has a non-compound example.

    "Every glass is ..."

The quantifier does not change the grammatical number of the subject.

I think that's the case for all the "every <noun>". "Every human is a person", for example. This would make sense, to put it in programming terms - the verb applies to an element in an array of people, not the array itself (which would be plural): for every single human, that human is a person.