You already know that first-order logic has a completeness theorem. That means that we can determine validity in first-order logic by looking at deductions - it makes proof theory possible. In second-order logic with full semantics, because there is no completeness theorem, to study things like validity we end up having to answer questions about the power set of the domain.
Here's an example. There is a sentence $\phi_1$ in the second-order language of ordered fields that characterizes the real numbers, up to isomorphism, in second-order logic with full semantics. There is another sentence $\phi_2$, in the language with just equality, which states that the domain has cardinality $\aleph_1$ (that is, any model of $\phi_2$ in second-order logic with full semantics has a domain of that cardinality). Now in order to show that $\phi_1 \to \phi_2$ in this logic, we would have to prove the continuum hypothesis, and to disprove that implication we would have to disprove the continuum hypothesis (this is because $\phi_1$ has only one model up to isomorphism).
Examples like this give us a sense that studying second-order logic with full semantics comes down, in many cases, to studying set theory. But if that's that case, many people say, why not just study set theory, as with ZFC? Studying set theory in the guise of "logic" only seems to obfuscate what's going on.
Moreover, for those who want to use the logic for foundational purposes, it is unattractive to pick a logic that seems to already have the answers to set-theoretic questions like the continuum hypothesis built into it - this goes against the idea that "logic" itself should make a minimal number of ontological assumptions.
This sort of argument was made in detail by Quine, who called second-order logic with full semantics "set theory in sheep's clothing". Not everyone agrees with this, and many people do use second-order logic with Henkin semantics as a way to keep the expressiveness without including the set theory. But the dominant opinion accepts Quine's argument.
I also recommend "The Road to Modern Logic-An Interpretation" by José Ferreirós, Bulletin of Symbolic Logic (2001), 441-484. This paper has a very nice historical study of the development of what is now called first-order logic.
The answer to your question is a qualified no. Part of the reason is that we can't assume that every object in our model is named by an individual constant. So for instance, it could be that our model satisfies the sentence $\bigwedge_{i \in I} \neg P(c_i)$, where "$\bigwedge_{i \in I}$" indicates (possibly infinite) conjunction over index set $I$, and $c_i$ are all constants of the language, and yet this same model also satisfies the sentence $\exists x P(x)$. It's just that the object in our model which satisfies $P(x)$ is unnamed.
Of course, you're right that there is a strong analogy between quantifiers and infinite conjunctions/disjunctions in the following sense: if we require that every object in our domain is named by a constant, and if we allow for arbitrary conjuncts/disjuncts, then we can translate the quantified sentences into quantifier-free sentences using (possibly infinite) conjunctions/disjunctions. Logicians sometimes define substitutional quantifiers for this purpose: for instance, letting $\Sigma$ be a new substitutional quantifier, $\Sigma x \varphi(x)$ is true in a model just in case for some constant $c$, $\varphi(c)$ is true in that model, i.e. just in case $\bigvee_{i \in I} \varphi(c_i)$ is true in that model, where $I$ indexes the constants of $L$.
With that said, an infinitary propositional logic without quantifiers is not the same as a first-order logic with quantifiers. For one thing, in a propositional logic, you can only say $p$ is true or false. Your models aren't collections of objects with structure, but rather are simply truth value assignments for the proposition letters. So it's hard to say in what sense, if any, an infinitary propositional logic is the same as first-order logic without infinitary conjunctions/disjunctions. Their models don't even look alike.
Furthermore, even an infinitary predicate logic without quantifiers fails to be equivalent to first-order logic with quantifiers (but only finite conjunctions/disjunctions). The reason is simple: in first-order logic, there is no sentence which is true exactly when the domain is infinite. However, if the language you invoke has (at least) countably many constants $c_i$, then the sentence $\bigwedge_{\substack{i,j \in \omega \\ i \neq j}} c_i \neq c_j$ can only be true in infinite models. Hence, infinitary predicate logic without quantifiers is not compact, and so can't be equivalent to first-order logic.
Best Answer
This system of quantified propositional logic is straightforward to interpret into first-order logic. We make a theory $T$ that has a single, unary relation symbol, say $P$, and no other symbols in the signature, not even equality. Then, to quantify over "propositional variables", we quantify over elements in first order logic as usual. For each element $x$ in a model of $T$, $Px$ is either true or false, so the elements of the model can be treated as if they were propositional variables.
Thus the quantified propositional sentence $(\exists Q)(\forall R)[R \lor Q]$ is interpreted into $T$ as $(\exists q)(\forall r)[Pr \lor Pq]$. In this way, every sentence of quantified propositional logic is interpreted as a sentence of $T$, and vice-versa.
If we wanted to add constant symbols to $T$, that would be equivalent to adding constant (i.e. non-variable) propositional variables to quantified propositional logic.
I would suggest that the main reason that we don't bother having quantified propositional variables in the "usual" framework for first-order logic is that they are not useful for formalizing the typical mathematical theories (group theory, linear orders, set theory, arithmetic, etc.), and a central goal in most presentations of first-order logic is to be able to formalize these theories. The same holds for $\lambda$ terms to define functions. There is no reason that they could not be included in first-order theories, and in fact they sometimes are, but most presentations have no use for them.