There are certain limitations to FOL, particularly the Lowenheim-Skolem theorem which is why we have to use HOL for models which are uncountably infinite because using an countably infinite number of sentences we can always construct a countable model. For very elementary definitions in Mathematics such as the least upper bound property for real numbers (or Dedekind cuts) we have to use second order logic. First order logic suffices for most of arithmetic, but mathematical induction is second order (omega incompleteness comes to mind), which we frequently use in arithmetic, which in turn is equivalent to the axiom of choice and the well ordering principle (which intuitionist's reject).
Having said that, first we have to address the question why any of us should be interested in any Symbolic Logic at all. Many professional mathematicians don't find symbolic logic interesting or useful either. Most of the time we use a metalanguage to just how that a proof exists in the object language by useful metalogical theorems and subsidiary deduction rules (Defined in Kleene, Stephen (1980). Introduction to meta-mathematics. North Holland. pp. 102–106. ISBN 9780720421033).
The primary reason we developed symbolic logic at all was to just concentrate on the syntax and not consider the semantics at all, do mechanical symbol shunting and yet be able to reason correctly, viz. soundness. One could argue that the motivation of developing symbolic logic was enabling a Turing machine to reason for us. David Hilbert had already shown that in Plane Geometry (Euclid) you need not understand what a point or a line means, but yet be able to prove correct theorems just by syntactic manipulation.
First order logic is philosophically interesting when it comes to understanding the limits of Turing machines against human cognition, because it exhibits both soundness and completeness. There has been much speculation in this problem, even by Kurt Godel himself, who gave the disjunction that either the mind is a machine or there exists infinitely many diophantine equations which cannot be solved, as a corollary of omega incompleteness of FOL. It's also handy when you are arguing or checking arguments. The short answer is, despite it's limitations, FOL is useful. We are perfectly aware of its limitations, and we are also aware that if we are to circumvent it's limitations, soundness and completeness have to be sacrificed. Whenever a certain argument is effable in either FOL or propositional logic, one should go with that, because it is much more reliable. I personally think, as Poincare opined that logic is good for checking things, but it's not useful for creating new things. There may be differences of opinions, but we already know that 3-SAT is NP-complete, so we have to wish ourselves luck in deriving semantically true statements using a computer. As far as the "ancestor" relationship goes in defining FOL, I do not see that as a problem. What I can say is simply using FOL and the compactness theorem that ∃ x ∀ n ∈ N x < 1/n, which was I believe what Leibniz argued both in his calculus and monadology, but was unable to prove. One of the consequences of this result is now the philosopher and the theoretical physicist has to consider infinitesimals in their science, metaphysics, and pataphysics.
In conclusion, philosophers are interested in FOL because there has been positive results in studying it by philosophers, model theorists, proof theorists and so on. There are some truth's, given we have defined our semantics, we can conclusively show which remains dubious in any metalanguage. It is alive and there are yet things to understand about it, and interpret about it.