I should perhaps first clarify how the traditional theory works:
Quantifiers with a slot for a subject restriction, like some, every, do not denote sets of sets. They denote binary relations between sets: [[some]] = the set of all pairs of sets F, G which overlap.
It is the quantifiers combined with a subject noun phrase that denote sets of sets: [[some men]] = the set of all sets G which intersect with the set of men. The quantified subject is formed by plugging in the subject noun phrase in for the first argument F in the quantifier relation.
The quantified sentence is then formed by plugging in the verb phrase for the second argument G: [[some men are mortal]] = true iff the set of mortals is among the set of sets which intersect with the set of men, i.e., true iff men and mortals intersect, and false otherwise.
Unrestricted quantifiers that can not accommodate another noun phrase and straightforwardly combine with a verb phrase, like something, someone, are of the same type as a restricted quantifier + subject restriction, they denote sets of sets.
In the computation of [[some men]], there is no explicit reference to the set of mortals. [[some men]] is simply the set of all sets which intersect with the men. It does not have to be established at this point which of these sets coincides with what is comprised by being mortal, or any predicate for that matter. The sets are anonymous. [[some men]] consists of any conceivable collection of individuals so long as it has at least one element with the men in common; these arbitrary sets exist independently of linguistic expressions that describe them, many of them may not even fit a single concise description at all. It is only when combining the quantified subject with the verb phrase in question that [[mortal]] has to be evaluated, in order to establish whether what it extends to coincides with one of those sets that intersect with the set of men, i.e., whether what man comprises and what mortal comprises have at least one element in common. But before putting together the subject and the verb phrase, [[mortal]] is nowhere needed in the definition of [[some men]].
If I am understanding you correctly, you are proposing that quantified determiner phrases like some men do not denote sets of sets, but sets of individuals, with the requirement that this set always be a subset of the subject noun phrase. Then a quantifier must be not a relation between sets (something where you put in two sets and get out yes or no), but a function from sets to sets (something where you put in a set and get out another set): some would take a set and return some subset of it.
What would this function look like? Which subset should be assigned to the set of men such that it fits the nature of "some"? And how should it combine with the verb phrase predicate? Which relation does the set of mortals need to have with which particular subset of the men?
If a subject with the quantifier incorporated denotes a set and a verb phrase denotes another set, what is the additional rule that relates them? With the traditional theory of generalized quantifiers, a quantifier combines with two predicates and gives you 'yes' or 'no', depending on whether or not the two sets stand in the specified relation, so that the sentence can be composed to have a truth value. If your quantifier no longer does the job of comparing two sets, and instead gives you some subset of the subject noun phrase and leaves you with two sets (the quantified subject and the verb phrase), what other hidden syntactic element makes sure that these two sets combine to something which can be true or false, and how?
And how would you go about reasonably defining a quantifier like no (as in No man is mortal) while demanding that no man be a subset of the set of men, when the meaning of no is precisely that whatever is predicated in the verb phrase does not share any elements with the set of men?
I see no easy way to define the standard quantifiers and the rules for their composition in this way.
I am also not convinced that such an assignment from sets to subsets would be more intuitive than the conventional theory of generalized quantifiers.
I agree that the conception of a set of sets, as the denotation for a half-baked sentence like some men with the subject noun phrase already plugged in but the verb phrase out of the picture, is hard to make sense of intuitively. Simply put, some men describes the properties which apply to some men; assuming that a property corresponds to a set of individuals, a quantified determiner phrase should thus be a set of such sets of individuals.
But it is probably easier not to overthink this intermediate composition step and instead plug in the two sets simultaneously: some denotes a relation between two sets, namely the relation of having a non-empty intersection. You plug in the men and the mortals, and get out 'yes' if they overlap, and 'no' otherwise. all denotes a relation between two sets, namely the relation of the one set being a subset of the other. You plug in the men and the mortals, and get out 'yes' if everything that falls under the description of a man is also inside the mortal circle, and 'no' otherwise.
A quantifier with two slots compares two sets. A quantified determiner phrase with the quantifier combined with the subject collects which properties apply to the subject in the specified way. A quantified sentence with the quantifier combined with the subject and the verb phrase has a truth value, depending on whether the subject and the verb phrase stand in the specified relation. I think this makes perfect sense intuitively.