Your argument has nothing to do with well-orders. For example, we can also prove that for every real number $x$, the equation $x=x$ holds. That is, we proved $\forall x\in\mathbb{R}\ x=x$. Your suggestion is now to consider every particular real number $a$, and realize that we have a proof that $a=a$. Does this give uncountably many proofs, one for each real number? No, because $a=a$ is not a statement in the formal language, which you said is finite. What is happening is that we have expanded the language by allowing explicit reference to $a$, and this makes your language uncountable.

Similarly, for the proper class case, we can prove $\forall x\ x=x$, where the universal quantifier ranges over all objects $x$. Does this give a proper class of proofs, of the statements $a=a$ for every object $a$? No, because those statements are not in the formal language. 

So ultimately, the solution to your confusion is to pay more attention to what exactly is in the formal language and what is not. If we can refer to the objects directly with constants in the formal language, then the claim that the language has a finite alphabet is false, invalidating the first part of the argument that there are only countably many proofs. And if we cannot refer to those objects with constants in the formal language, then we cannot have proofs of the statement $a=a$ or any other statement about $a$, since these assertions are not in the formal language.