Yes, that perception is correct.
The reason (why) that perception is correct is that why is a rather special relative pronoun.
Indeed, it's a pronoun that can only refer to one word: reason.
Try it with anything else and you get garbage:
- the reason why he did it
- *the cause why he did it
- *the intention why he did it
- *the effect why he did it
- *the thing why he did it
(ungrammatical phrases and sentences are marked with an asterisk)
and these are nouns that could make sense with a Why relative pronoun;
try it, if you dare, with nouns that couldn't, like rock, salamander, or durability.
Relative why can be freely substituted with that, like any restrictive relative marker.
I.e, substituting that for why in the sentences above produces exactly the same pattern of grammaticality and ungrammaticality:
- the reason that he did it
- *the cause that he did it
- *the intention that he did it
- *the effect that he did it
- *the thing that he did it
More important, why refers to an adverbial clause or phrase of some sort in the relative clause -- certainly not a noun phrase -- and therefore it can't possibly be the subject of the clause. This is important because relative pronouns that are the subject of their relative clause (like the man who/that came to dinner) cannot be deleted. But adverbial wh-words -- like why, where, when, and sometimes how -- can't ever fall into that category.
This means why -- or that -- can be freely deleted after reason. I.e, deleting why in the sentences above also produces exactly the same pattern of grammaticality and ungrammaticality.
It's not a matter of redundancy; all pronouns are redundant, after all.
It's just that why is very limited in its distribution.
Not quite as limited as how, however.
How can't be used at all as a relative pronoun;
one may use that, or nothing at all, but how (which refers to way) is ungrammatical as a relative marker.
- the way that he did it
- the way he did it
- *the way how he did it