In my view, both are possible. Perhaps 'we' is preferred because of the similarity between 'as' and 'us' in pronunciation.
The same form of the pronoun 'we' (often called 'subjective case') is used for pronouns introducing a clause (that is, functioning as a subject within the main sentence). Eg: In "Not everyone eats as we (eat)" 'we' is the subject corresponding to the verb 'eat' ('as we eat' is a clause). In a case like this, it is more likely that the verb (in this case 'eat') is omitted or replaced by 'do'.
"Not everyone eats as we do"
Subjective case of pronouns:
Objective case of pronouns
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