$$ (\forall x \in \mathbb{N})[x < 3 \implies x \ge 3] \lor (\forall x \in \mathbb{N})[x \ge 3 \implies x < 3] $$ is not true, and your analysis doesn't imply that it ought to be.

What is true by your argument is $$ (\forall x \in \mathbb{N})\bigl[(x < 3 \implies x \ge 3) \lor (x \ge 3 \implies x < 3)\bigr] $$ but you can't move the quantifiers into the middle of its structure and expect it to remain true.

In general, $(A\to B)\lor(B\to A)$ is always true in classical logic, no matter whether $A$ and $B$ are negations of each other or not. Perhaps even more intuition-challenging, $(A\to B)\lor(B\to C)$ is always true too.

$$ (\forall x \in \mathbb{N})[x < 3 \implies x \ge 3] \lor (\forall x \in \mathbb{N})[x \ge 3 \implies x < 3] $$ is not true, and your analysis doesn't imply that it ought to be.

What is true by your argument is $$ (\forall x \in \mathbb{N})\bigl[(x < 3 \implies x \ge 3) \lor (x \ge 3 \implies x < 3)\bigr] $$ but you can't move the quantifiers into the middle of its structure and expect it to remain true.

In general, $(A\to B)\lor(B\to A)$ is always true in classical logic, no matter whether $A$ and $B$ are negations of each other or not. Perhaps even more intuition-challenging, $(A\to B)\lor(B\to C)$ is always true too.

Response to edit: I think you're being confused by two different uses of "if ... then" in mathematical English.

1. On one hand in casual non-formal mathematical English, "if it's heads, then it's tails" would mean something like

In every relevant situation where it's heads, it is also the case that it's tails.

(and one then hopes that the context makes it clear which situations are "relevant").

2. On the other hand, in (classical) formal logic, the formula "${\rm heads}\Rightarrow{\rm tails}$" means

In the particular situatation we're looking at, it is not the case that it is heads yet it isn't tails.

Which is the same as saying

The particular situation we're looking at is not a counterexample to sense (1).

Confusion creeps in because "${\rm heads}\Rightarrow{\rm tails}$" is often pronounced "if heads then tails", as if its meaning were the same as in (1), but it is not when formulas are involved.

The sense in which you can prove that $(A\Rightarrow \neg A)\lor(\neg A\Rightarrow A)$ is true is (2) and only (2). It stops being true if you try to reinterpret the $\Rightarrow$s as if they meant (1).

Furthermore it it looks like you're also confusing yourself by the difference between "is true" and "has a proof" when you say that "this or that" is true but "this" and "that" are both false.

What you mean here is that you believe you have a proof of "this or that". But clearly it is not true because the definition of "or" is that "this or that" is true exactly when at least one of "this" and "that" is true -- and you have just argued that they are both false.

So when you ask,

How can "this" be false and "that" be false and "this or that" be true?

the answer is that they can't and you're mistaken about at least one of those parts. Same as if you have an apparent proof that concludes $2+2=3$: you shouldn't be asking "why is $2+2$ not $4$?" but "what is wrong with this thing that looks like a proof but obviously can't be?"

$$ (\forall x \in \mathbb{N})[x < 3 \implies x \ge 3] \lor (\forall x \in \mathbb{N})[x \ge 3 \implies x < 3] $$ is not true, and your analysis doesn't imply that it ought to be.

What is true by your argument is $$ (\forall x \in \mathbb{N})\bigl[(x < 3 \implies x \ge 3) \lor (x \ge 3 \implies x < 3)\bigr] $$ but you can't move the quantifiers into the middle of its structure and expect it to remain true.

In general, $(A\to B)\lor(B\to A)$ is always true in classical logic, no matter whether $A$ and $B$ are each other's negations or not. Perhaps even more intuition-challenging, $(A\to B)\lor(B\to C)$ is always true too.

$$ (\forall x \in \mathbb{N})[x < 3 \implies x \ge 3] \lor (\forall x \in \mathbb{N})[x \ge 3 \implies x < 3] $$ is not true, and your analysis doesn't imply that it ought to be.

What is true by your argument is $$ (\forall x \in \mathbb{N})\bigl[(x < 3 \implies x \ge 3) \lor (x \ge 3 \implies x < 3)\bigr] $$ but you can't move the quantifiers into the middle of its structure and expect it to remain true.

In general, $(A\to B)\lor(B\to A)$ is always true in classical logic, no matter whether $A$ and $B$ are negations of each other or not. Perhaps even more intuition-challenging, $(A\to B)\lor(B\to C)$ is always true too.