Irving Copi calls propositions such as "James' son is a man" or "Socrates is mortal" singular propositions. They are nonstandard-form propositions. They need to be translated into standard form categorical propositions which relate classes before being used in categorical syllogisms. He recommends the following: (page 239)
To every individual object there corresponds a unique unit class (one-membered class) whose only member is that object itself. Then to assert that an object s belongs to a class P is logically equivalent to asserting that the unit class S containing just that object s is wholly included in the class P.
Copi quotes Kant (Critique of Pure Reason, trans. N.K. Smith, p. 107) to justify such a translation: (page 240)
Logicians are justified in saying that, in the employment of judgments in syllogism, singular judgments can be treated like those that are universal.
Treating singular propositions as universal works for Barbara (AAA-1). The issue of existential import does not arise because all of the propositions are universal. It does not work, however, for other forms, such as, Darapi AAI-3 where two universal propositions have a particular conclusion. This breaks Copi's rule prohibiting premises without existential import deducing a conclusion which does have existential import: (page 240)
s is M goes into the invalid All S is M.
s is H AAI-3 categorical syllogism All S is M.
∴ Some H is M. ∴ Some H is M.
Copi also notes that Bertrand Russell viewed clarifying the status of these singular propositions as the first of two advances made by modern logic over Aristotelian term logic: (page 66)
The first advance consisted in separating propositions of the form 'Socrates is mortal' from propositions of the form 'All Greeks are mortal'. In Aristotle and in the accepted doctrine of the syllogism (which Kant thought forever incapable of improvement), these two forms of proposition are treated as indistinguishable or, at any rate, as not differing in any important way. But, in fact, neither logic nor arithmetic can get far until the two forms have been seen to be completely different. 'Socrates is mortal' attributes a predicate to a subject which is named. 'All Greeks are mortal' expresses a relation of two predicates - viz. 'Greek' and 'mortal'.
Both Copi and Russell, although perhaps not Kant or Aristotle, would have agreed with the OP about the need to treat these singular propositions more carefully.
Copi, I. M. Introduction to Logic. Sixth Edition. (1982) Macmillan.
Russell, B. My Philosophical Development. (1959) George Allen & Unwin. Internet Archive: https://archive.org/details/myphilosophicald0000russ/page/n6