Modal verb should not redirect to English modal auxiliary verb because other languages have modal verbs.
From the en.wikipedia.org
It doesn't matter how these sources use the terms modal, modal verb, modal auxiliary, or auxiliary.
From the en.wikipedia.org
In the English language, a modal verb is a type of auxiliary verb.
From the en.wikipedia.org
If the verbal expression of modality involves the use of an auxiliary verb, that auxiliary is called a modal verb.
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A verb phrase contains verbs which can be lexical, auxiliary, or modal.
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Modal auxiliary verbs give more information about the function of the main verb that follows it.
From the en.wikipedia.org
In English, main verbs but not modal verbs always require the auxiliary verb do to form negations and questions, and do can be used with main verbs to form emphatic affirmative statements.
From the en.wikipedia.org
It's just that these moods aren't technically considered to exist in English, since they are usually indicated by modal auxiliary verbs rather than by separate verb forms.