I'll attempt an answer.
The key issue is the contrast between lay intuition, and mathematical logic. Lay intuition tells us that nouns and verbs are the most 'contentful' parts of a sentence. Mathematical logic brings its rules for operators and their scopes.
If we have the expression
∃x(F(x)), we can't have the
x scoping wider than the outermost parentheses here. If an
x had some sort of a universal scope, for example, over
G too in
G(x)∧∃x(F(x)), we'd soon have a formalism that only talks about constant terms in the universe. Such
xs would be useless as variables.
Now, note that the logical terms that we introduce for describing sentences are based on the principle of compositionality. If we had used
"man" => λx.man(x), and
"some" => λF∃x.F(x) to derive
"some man" => ∃x.man(x), this would not correspond to what we understand of the phrase "some man". Instead, it would correspond to "there is a man".
The proper expression for "some" would be
"some man" => λQ∃x.man(x)∧Q(x), and (after a bit of α-conversion to avoid name clashes)
"some woman" => λR∃y.woman(y)∧R(y).
Here are two more pieces of the puzzle:
"visit" => λSλu.S(λv(visit(v,u))), and
"and" => λHλJ.H∧J.
We now have NPs that have become functions that accept Vs and VPs as arguments. Determiners, that play a subordinate role in Phrase Structures, play a key role here in controlling the scope of variables, and the order in which functions apply over arguments.
Given this, if you were to apply the object over the verb, and then apply the subject over the resultant, you'd have the man and the woman each visiting a garden, that may or may not be the same garden. However, if you were to apply the subject over the verb, and then apply the object over the resultant, you'd have them both visiting the same garden.
In all this, we haven't brought in (Neo-)Davidsonian Event semantics, so we have made no commitments at all about whether or not the visit(s) to the garden(s) happened at the same time.