Is a child's saliva and spittle rendered impure after vomiting due to its mixture with impurity?

Ibn al-Qayyim holds that the saliva and spittle of an infant fall under matters of widespread affliction, and the Lawgiver knew that a child often vomits and that it is not possible to wash his mouth. His saliva also flows onto those who raise him. Yet, the Lawgiver did not command the washing of clothes from this, nor did He prohibit prayer in them. For this reason, a group of jurists stated that this is a type of impurity that is overlooked due to hardship and necessity. In fact, a child's saliva purifies his mouth out of necessity. They supported this with what was reported from Abu Qatadah that the Prophet, peace and blessings be upon him, would tilt a vessel towards a cat until it drank, and then he would perform ablution with the remaining water.