can we call these logical connectives the propositional functions?
The issue is legitimate, but "history" followed a different path and now we have a quite established convention.
For the source, see :
- Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (1st ed.1910, 2nd ed.1927); see Introduction (to the 1st edition), page 1-on:
[page 5] Variables will be denoted by single letters [...] $p, q, r$ will be called propositional letters, and will stand for variable propositions [sic! meaning that they are variables for propositions].
[page 6] An aggregation of propositions [...] into a single proposition more complex than its constituents, is a function with propositions as arguments. [...] there are four special cases which are of fundamental importance, since all the aggregations of subordinate propositions into one complex proposition which occur in the sequel are formed out of them step by step. They are (1) the Contradictory Function, (2) the Logical Sum, or Disjunctive Function, (3) the Logical Product, or Conjunctive Function, (4) the Implicative Function. [...]
The Contradictory Function with argument $p$, where $p$ is any proposition,
is the proposition which is the contradictory of $p$, that is, the proposition
asserting that $p$ is not true. This is denoted by $\sim p$.
[page 7] These four functions of propositions are the fundamental constant (i.e.definite) propositional functions with propositions as arguments, and all other constant propositional functions with propositions as arguments [...] are formed out of them by successive steps.
[page 14] Let $\phi x$ be a statement containing a variable $x$ and such that it becomes a proposition when $x$ is given any fixed determined meaning. Then $\phi x$ is called a "propositional function"; it is not a proposition, since owing to the ambiguity of $x$ it really makes no assertion at all.
Principia Mathematica has been blamed for this kind of sloppiness regarding the sintactical specifications of the language.
Thus, the two different usages of "propositional function" has been subsequently replaced by connective and open formula respectively.