The Cambridge Grammar of the English Language consistently uses the term "auxiliary (verb)" to refer to be, have, do, will/can/may/must, etc., but CGEL doesn't treat auxiliaries as mere assistants (i.e., auxiliaries) of lexical verbs since CGEL adopts the catenative-auxiliary analysis over the dependent-auxiliary analysis.
The former analysis treats "auxiliaries" as the heads of the VPs whereas the latter analysis treats them as dependents of the following lexical verb.
Why would CGEL keep the term "auxiliary (verb)", which I think is incongruous with the catenative-auxiliary analysis?
Is there an alternative grammar term for 'auxiliary (verb)' that is in use in a modern grammar that adopts the catenative-auxiliary analysis?
This question arose out of an earlier answer to the question Is “Helper Verb” Old School?