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I think you got that wrong, "valedixit" is just third person perfect, "he/she/it said goodbye", the verb suffix does not encode gender.

That is true. It could also be used by a "he", so there is a bit of ambiguity.

I'm really curious, where is the gendered information in "valedixit"? I feel like the proposed translation misses that romantic weight by keeping the gender ambiguous. What am I missing?

I don't speak Latin, but according to my short research (aka googling), it does not distinguish between third-person masculine and feminine. So the ambiguity is also present in Latin.

The thing missing is the context in which the sentence is used.

From the comments on stackexchange: "if the context refers to this person enough to make it clear who "she" is, it should also make it clear who "he/she/it/they" is."

That makes a lot of sense, thank you.

it is not encoded in the verb ending.

How very postmodern.