Healing can’t be an ordinary weave.
weaves do a thing. A construct acts situationally. If someone is anemic, it increases their blood clot factor, if someone is suffering clots it reduces the blood clot factor. If someone has a wound in one part of the body, it only “heals” there.
we were introduced to the idea of other kinds of weaves that heal specific parts of the body like the heart, and one Aes Sedai can dial in the power so carefully that she can heal someone just enough to live without healing the body so completely that the patient dies from the loss of energy, but she can’t prevent the weaves from healing a scratch while it heals the lung.