To what end would you ever assert ‘p∨q’? Why not instead find out which is true and merely assert that instead? The main reason is that there are unspoken variables in both ‘p’ and ‘q’. Here is a practical use of ∨:
A child is either a boy or a girl.More formally, for every child x, (x is a boy)∨(x is a girl). Here the disjunction serves a real and useful purpose. It becomes useful in context of universal quantifiers.
Similarly p→q is useful in such contexts. “All men are mortal.” is formally ∀x(man(x) → mortal(x)).
The sentential calculus, from the perspective of predicate calculus lacks quantifiers. When a theorem from sentential calculus is imported into predicate calculus it is necessary to ‘close’ it by adding enough universal quantifiers to the whole theorem to eliminate all the free individual variables. This is a convention that is introduced to formal logic well into the development of the subject. It is a good and useful convention, sometimes left unsaid in even mathematical contexts; but it is only a convention. The man on the street cannot be expected to know this and is justified in unconsciously interpreting (p→q)∨(q→p) as (∀x(p→q))∨(∀x(q→p)) rather than the mathematically conventional ∀x((p→q)∨(q→p)). The later is true but the former is not generally true.
A closely related form of notational confusion is to confuse the proposition with the predicate and thus to see p→q as a relation between two predicates indeed embodied as sets. In such a frame of mind p→q is taken to mean what is mathematically expressed as P⊆Q where P is {x|p} and p is imagined to somehow name x. (p→q)∨(q→p) thus seems to mean (P⊆Q)∨(Q⊆P) which is not generally so.
In short people just don’t know the notation which is not surprising psychologically or philosophically.