Consider the empty language $L = \{\}$. Then we are said to be able to define the set of infinitely many elements, say $K = \{x_1,\cdots,x_n,\cdots\}$, as the set satisfying the infinite set of finite formulas $\{ \phi_1,\cdots,\phi_n,\cdots \}$ where
$$\phi_n := \exists_{\{x_1,\cdots,x_n\}} \bigwedge_{\{i, j: i \neq j\}}(x_i \neq x_j)$$
But this statement is equivalent to the infinite formula
$$ \exists_{\{x_1, \cdots, x_n, \cdots\}} \bigwedge_{\{i, j: i \neq j\}}(x_i \neq x_j).$$
I claim that every infinite formula can be equivalently expressed as an infinite set of finite formulas.
Thus I do not see why do we disallow the second but not the first?
Does disallowing infinite sets of finite formulas have a big impact on first order logic?
The language $L_{\infty\omega}$ allows infinite disjunctions and conjunctions. It doesn’t allow infinitely deep chains of quantifiers, but you can easily rewrite your “infinitely many elements” example to abide by this restriction.
$L_{\infty\omega}$ has been extensively studied, and has interesting properties, but not the same properties as first-order logic. For example, the compactness theorem (perhaps the central tool in model theory) fails.
This is easy to see. The negation of the sentence “there are infinitely many elements” is also a formula of $L_{\infty\omega}$. Put this negation together with the infinite family of sentences “there are at least $n$ elements”, and you have a theory whose finite subtheories are all satisfiable, but which is not satisfiable.
So there are good reasons to study first-order theories, and not just $L_{\infty\omega}$.
Infinite conjunctions of sentences are equivalent to infinite families of sentences, as you’ve noticed. Not so infinite disjunctions. For example, “there are only finitely many elements” is easily expressed as an infinite disjunction, but cannot be expressed by an infinite family of sentences of first-order logic.
The notion of “type” in a first-order language serves as a quasi-substitute for an infinite conjunction, in some cases. The omitting types theorem can be regarded as saying something about satisfying an infinite disjunction.