I think there's also a level of symbolism to the sword that wouldn't really work for Theresa's role. It's not only too violent, but too "immediate". It strikes down that in front of you but goes no further.
Look at the iconic sword users in the main plot: Theresis, Talulah, Ch'en, and—on specific occasions—Amiya. Theresis and Talulah are both characters trying to proceed their plans through direct military action, a raw application of force for a single campaign. Ch'en wavers, but only truly becomes the swordmaster when she leaves the LGD and the politics of Lungmen behind and seeks to handle things personally, tackling only that which she can deal with using her own two hands.
Amiya isn't usually a sword user—most of the time, she's a Caster like Theresa. The only times we see the sword come out are the times where she's at the limit, and violence is the last resort. She couldn't find any other path to stop Kaschey, and the force of the Ursus Army was otherwise overwhelming. The sword itself comes from Qui'lon, who is best known for being the Terran Buddha... but the sword represents the moment he sank into a great deal of killing. It doesn't represent a long-term plan, but action in the moment.
I will also note Siege, who is trapped in an ambivalent state. She has the sword, but doesn't really use it. She recognises it as a symbol and detests it, but can't be rid of it either. For a time she holds the sword and leads the Exemplars into battle, but in the end she hands it to the people of Londinium for them to use instead. She ends her arc as a politician, not a fighter; while violence might have solved the immediate problem, it won't help solve Londinium's longer-term problems.
So, Theresa. She represents a beautiful far-off dream for the Sarkaz, that one day they won't need violence. Our first sight of her on the battlefield is as a radiant beacon drowning out the fighting. She stands in a stainless white dress, and she wields no weapon. If she took up the sword, it would be a visible sign that the Sarkaz's saviour could do no better than the rest, that bloodshed was always going to be the path forwards. But it would also narrow down her vision—she's not fighting for the future, she's only fighting for the present moment.
(Of course she was still embroiled in a civil war—but she also seemed above the conflict. The actual fighting was done by her Knights, so to speak, a pattern that started with Theresis. W was captivated by her bending over to fix an automatic door, not by her snuffing out a life.)
To some extent this is just repeating the analysis in the first paragraph of OP. They are trying to build her up as this mythical, ethereal figure, the elegant queen in her tower that flies above the messiness of the battlefield. Having her wield the blade herself would despoil that image. Her aura farming comes from rewriting the world with calm composure and steel will, not from doing Vergil's Motivated combo. We've seen her ruthlessness when it really matters.
All that said, we do have that one image of Theresa holding a sword... at the very moment of her failure. She bloodies her own hands, forces the crown onto Amiya, and dies. She lost her vision of the future, and could only act in the moment to hope someone else might make something beautiful. Babel's dream shatters. The Sarkaz don't just lose their saviour physically, the ideal she represents is also tarnished—war dragged her down, war claimed her. It's a moment that says, maybe there really is no way out of this mire.
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