Is it correct to say that the Prophet (peace be upon him) allowed women to trail their head coverings on the ground, despite what that might cause in terms of collecting dirt and invalidating prayers?

The society of the Companions was keen on protecting and veiling women. A woman's entire body is considered awrah (private parts that must be covered), and she must cover all of her body when going out. For this reason, the female Companions would lengthen their garments to cover their feet. However, at home, a woman is not obliged to do so.

The soiling of a garment due to its length is insignificant compared to protecting women and closing the door to fitna (temptation/strife). The fundamental principle for women is to remain in their homes, and they should not go out except when necessary.

Regarding the soiling of the garment and perhaps its contamination with impurity, it was reported from Umm Salamah that the Prophet, peace and blessings be upon him, said concerning the hem of a garment that passes through a dirty and then a clean place: "What comes after it purifies it."