I think the problem is that the meaning of $P \Rightarrow Q$ in logic doesn't correspond precisely with the meaning of "if ... then ..." in natural language.
Natural language sentences of the form "if ... then ..." are usually understood to be general statements. For example, "if it's raining then I take an umbrella" would be understood as a statement about my behaviour in general, not my behaviour on one particular day. So if we were to translate it into formal logic, it should be something like:
$$\forall d : R(d) \Rightarrow U(d)$$
where $R(d)$ means it's raining on day $d$, and $U(d)$ means I take an umbrella on day $d$. Note that it's the universal quantifier $\forall$, not the use of $\Rightarrow$, that makes this a general statement.
Now suppose my statement that "if it's raining, then I take an umbrella" is negated. This would mean that sometimes, it's raining but I don't take an umbrella. That would be written in formal logic as follows:
$$\exists d : R(d) \land \neg U(d)$$
In formal logic, $\neg \forall x : \phi$ is equivalent to $\exists x : \neg \phi$. Our intuition is that the second quantified expression above should be the negation of the first; so for that intuition to hold formally, it must be the case that $R(d) \land \neg U(d)$ is equivalent to $\neg\big(R(d) \Rightarrow U(d)\big)$.
If you read $P \Rightarrow Q$ as meaning "if P then Q" then your intuition can easily lead you astray. For example, the formula $(P \Rightarrow Q) \lor (Q \Rightarrow R)$ is a tautology. But if we apply our intuition, then a sentence like "either (if you play golf then you're the king) or (if you're the king then you live in space)" is clearly not a tautology, because not every golfer is the king, and the king doesn't live in space.
On the other hand, if you translate "if P then Q" as $\forall x : P(x) \Rightarrow Q(x)$ then the above statement would be more like:
$$\big( \forall x : P(x) \Rightarrow Q(x) \big) \lor \big( \forall x : Q(x) \Rightarrow R(x) \big)$$
which is indeed not a tautology in formal logic. So the conclusion is, it's often untenable to interpret $\Rightarrow$ as a general statement of the form "if ... then ...", because $\Rightarrow$ alone doesn't make it a general statement.