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I've now found where McCawley discusses the question (Section 6.3).
He argues that universal-quantifiers-as-expressed-in-English can
be analysed as having existential import, but needn't be and
perhaps even shouldn't be:
"Every member who pays all his bills on time is entitled to a 10%
discount"
-- would normally be understood to apply to members with no bills
to pay.
However, he also points out that there is some kind of conventional
implicature ("pragmatic presupposition") that restricts the
range of quantification to relevant candidates, where relevance
is contextually determined:
"Everyone who loves all their children is saintly"
-- does not claim that everyone childless is saintly.
--And.