Suppose we have the following: $\forall x \in X, P$.
Considering $x$ does not occur on the right hand side, we can also reformulate to: $X \rightarrow P$.
Let us now assume that $X$ means "it's raining" and $P$ "it's cold". We can read the second part as follows: "If it's raining, it's cold."
What is a reasonable formulation in English of the very same thing using the universally quantified version? "Forall whatever that's raining, it's cold." doesn't quite flow off the tongue.
You're right that an implication in natural language often it best interpreted as something universally quantified. Something like:
Your particular formalization of this intuition is problematic, though. If $X$ stands for "it's raining" (such that you can write $X\to P$), then $X$ is not a set, and therefore writing $x\in X$ is not meaningful.
If we want to make the implicit quantification explicit, then we need to parameterize both $X$ and $P$ to for example $X(t)$ and $P(t)$, meaning "at time $t$ it is raining (respectively, it is cold)". Then we can write $$(\forall t)(X(t)\to P(t)) $$
In some situations -- in particular in propositional logic as developed by introductory textbooks -- one can get away with keeping the quantification implicit, with the understanding that everything we say is assumed to hold at an arbitrary point in time.
This should not be sneered at -- technically, in proof theory, there's almost always an implicit quantification over all models of our theory. But it shouldn't lead us to think that keeping it implicit will always work for modeling natural-language statements. Depending in how complex the idea we want to express is, we may need to go to predictate logic or some kind of modal logic.
