"And the girl’s father shall say [to the elders]: [The father, but not the mother.] This teaches us that a woman is not permitted to speak in the presence of her husband [when others are present]. — [Sifrei 22:91] ואמר אבי הנערה: מלמד שאין רשות לאשה לדבר בפני האיש:"
Above is the OP's version of the Rashi in question.
First of all (even though this entry comes from Chabad.org; and I am from Chabad) this version of Rashi and its translation plus bracketed comments is misleading and mistaken IMHO.
Lets stick with the simple text of Rashi and then examine it.
"And the girl’s father shall say:" This teaches us that the woman is not permitted to speak "before" the man. (Sifrei 235)
Rashi actually uses no words of his own. He simply picks some words of the verse to comment on; and then quotes the Sifrei verbatim, without any explanations.
What does it mean?
The Sifsei Chachamim (note 100) comments on Rashi as follows:
1) The wife/mother is also standing here in the court. Yet, only the father is speaking to the court to verbalize his claim. This teaches us that "the woman is not permitted to speak "before" the man."
2) The Sifrei means that the girl should not be the one to lay out her claim against her husband's slander. Otherwise, the verse should have said that the girl speaks to the court etc.
The Mizrachi adds to #2 that we would expect the accused girl to be the one to respond in court here (not her proxy: father/mother). Since the Torah specifically shows the father speaking, we see the proof for the Sifrei that it is inappropriate for her to lay out her own defense and accusations against her estranged husband.
The Ibn Ezra adds that her parents stand for her if they are alive. If not, the court appoints a representative for her.
Now going with the #2 approach, the Sifrei may not be saying anything about women speaking or not speaking in front of their husbands at all? It could be that since this young girl might eventually go back to her husband, the less she fights with him in open court, the easier their reconciliation will be? Maybe the Sifrei is merely echoing the general Talmudic concept, that women (especially a young girl) should be allowed to avoid court proceedings as much as possible because of "tznius" (modesty). Perhaps she is simply being shielded from having to face her abusive husband directly? Perhaps all three? I have not found proof to anything specific here, but it does not prove a code of silence being imposed upon wives in front of their husbands in Halachah.
Even if we take the #1 explanation of the Sifsei Chachamim above (which excludes the mother from speaking before her husband) we still have no proof that women in general, may not speak in front of their husbands.
The Ramban explains that the reason the father speaks exclusively here, is because the monetary fine belongs to him should he win. This is something the mother has no claim to. Therefore, court etiquette has the father presenting the claim. The mother, would not be expected to appear in court in any case. However, since the mother is assumed to have been the parent who was highly involved in the wedding proceedings, she would have been the one to preserve the bed sheets in her possession. Therefore she is here in court and is involved in producing the evidence.
So, the Ramban has no code of Halachah here that states women may not speak in front of their husbands. However, he also does not even quote the Sifrei or its conclusion. Rashi does.
So, if Rashi did make his comment in line with the Sifsei Chachamim's #1 (excluding the mother from speaking), then does Rashi hold that women may not speak in front of their husbands Halachically?
Sefaria.org also translated Rashi with its own bracketed comments, as opposed to the explanation proposed by Chabad.org:
"... this teaches that a woman is not allowed to speak in the presence of her husband (if he, too, is concerned in the matter)."
This implies that if he is not concerned, she may speak in front of him.
Sefaria's version of the Sifrei actually has the following alternate reading:
ואמר אבי הנערה אל הזקנים. מכאן שאין רשות לאשה לדבר במקום האיש
"This teaches that a woman is not allowed to speak in "place" (meaning instead of) of her husband.
If we accept this understanding of the Sifrei, then the only time a wife should not speak before her husband is in a special case like a court of law, where she is supposed to be present, but her husband as head of household is the one to be the leader and lay out a defense on behalf of the family in front of the judges.
However, she certainly may speak to her husband in front of others and to others in front of her husband. Why can't she be social at a dinner party?
Even the Torah Temimah can accept this distinction. He compares this Sifrei to the idea in the Rambam, that a wife should treat her husband like a king/officer in charge. That would apply when there is an official spokesperson for the family (husband). She would not "but in" ahead of him, and speak in his stead.
This would be similar to Rashi's comment elsewhere on Genesis 24:50.
"And Laban and Bethuel answered and said, "The matter has emanated from the Lord. We cannot speak to you either bad or good."
Rashi: "And Laban and Bethuel answered: He [Laban] was wicked and jumped to reply before his father."
Finally, the Lubavitcher Rebbe in his discourses on Rashi/Chumash, says that Rashi's purpose in his commentary, is to answer the question of a five year old child who sees something amiss in a Torah verse.
In this case, the child could easily ask us why the Torah has both parents coming and then only has the father speak? OR ask us why the girl is not speaking up for herself like any other case?
Therefore Rashi searched through the words of our Talmud, and provided an answer that solves the child's problem. (This Sifrei, which shows us that the Torah chooses a situation where the parents must come together, but only the father speaks, answers that it is in order to teach us the lesson: 1) A wife shouldn't speak instead of her husband when addressing the court on behalf of the family OR 2) The young girl should not have to personally be directly involved if at all possible.
However, Rashi will pick the opinion in Talmudic literature that best answers the 5 year old's question. This is regardless of whether or not that opinion is the Halachically accepted one.
