I assume that Sansa and Arya are aware of Ned's wise words.
The Winterfell plot line this season has been centered around Littlefinger sowing discord amongst the Stark siblings.
However, Littlefinger's final scene shows Arya, Sansa, and Bran coming together to pass judgment on Lord Baelish.
Over the past several episodes all 3 siblings had touched the dagger.
They've held it and passed it between each other.
So while Sansa might have said the words and Arya's hand was on the blade when it killed Littlefinger they operated together as a unit.
The Starks did pass the sentence and the Starks swung the sword.
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Sansa was never taught this.
This lesson was likely never taught to Sansa or Arya to begin with.
In the original scene, the lesson was for the benefit of the male children present (Robb, Jon, Bran).
Given the attitudes towards what was proper for women to learn, he would not have bothered teaching this bit of wisdom to either Sansa or Arya (though Arya might have heard about it secondhand).
Sansa would have no reason to want to strike the blow herself.
In addition, she has been learning, the hard way, the art of manipulation.
It would fit with both this change in her personality, as well as her upbringing as a lady to let another hand do the dirty work.
She may have become aware of this later, but not likely - it's not the sort of thing that would come up in conversation before she was spirited away by Littlefinger, and it certainly was not among the behaviors he was trying to teach her.
If she has not had occasion to pass this sentence on someone, there is zero reason for anyone to mention it.
Certainly no one had the temerity to bring up at that moment.
Think of where she began, and where she is now.
That sort of thinking belongs to the bright, honorable world she was expecting long ago when Robert and friends first came to town, a world view that has long since been shattered.
Even if she IS aware of the tradition, it belongs to a world that must appear to her as naive and fanciful, where people do the right and honorable thing.
she's seen what happens to people that think that way.
Whether she knows of it or not, she has no reason to do the deed herself.
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As you say, the old way was The man who passes the sentence should swing the sword (emphasis mine).
There is no natural law of the universe that prevents someone else carrying out the sentence, which is what happened in this case.
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There is two quotes at work here.
"
The Man who passes the sentence should swing the sword" and "
The lone wolf dies, but the pack survives."
The event in question symbolizes that the Stark children aren't lone wolves anymore, but they are a pack again.
And the "man" (or rather the entity) that passes the sentence is the wolf pack.
Thus Bran, Sansa, and Arya are all one in respect to the first quote.
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I like to think that Arya added Baelish to her list once Bran told her that he was responsible for their father's death (assuming this happened behind the scenes) and therefore wanted to be the one to end his life.
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In the Season 7 finale of Game of Thrones,  The Dragon and the Wolf    Petyr Baelish is sentenced to death by Sansa, and Arya kills him   According to the old way,  The man who passes the sentence should swing the sword     So, why did the person who passed the sentence not carry it out?
Why was the person who passed the sentence in the Season 7 finale not also the executioner?
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There are a number of possible explanations for this, but it is difficult to say for sure which is the correct reason. Firstly, it could be considered that The Starks operated as a unit both when passing the sentence and then carrying it out. Alternatively, it could also be that Sansa was never taught the rule of needing to carry out the sentence. It's also possible that she was aware of the Old Way, but chose to ignore it because she doesn't see it as important. Views vary - Arya can be seen as striking the blow for the Starks as whole, as getting revenge for the death of her father or as simply ignorant of a rule that would - in such a society - have only been taught to the menfolk. The rule was anyway only a matter of custom - she was not breaking any deeper natural or universal law.
Altho Arya struck the blow she was doing it as a Stark and the whole execution - including the sentencing - was a combined effort of the Starks. Sansa was never taught this rule probably because it was never felt that she - as a women - would need to implement it. The rule is just customary - there is no natural or universal law behind it and therefore ultimately no obligation to honor it. Arya wanted to be the one who killed Baelish as he was responsible for her father's death - so it was partly a revenge killing.