The same is true for the opposite as well; that is, the sight, sound, smell, taste, and touch of a man are obstacles for a woman to reach extinguishment according to AN 1.6-10/EA 9.8/T 792.
EA 9.8 says that the sight of a man doesn’t lead a woman to permanent peace; it fetters her to a “prison” (judging from the context of the passage, I think it means rebirth); there’s no liberation for her; it leads her to the cycle of “going and coming” (death and rebirth), this world and the next, the cycle of the “five destinies” (hell, the animal realm, the ghost realm, the human realm, and the heavenly realm) for a long time.
T 792 says that the sight of a man leads impurity to a woman, sinks her to the bottomless pit of delusion and misery; it prevents her from reaching the supreme sanctuary. The said discourse also says that the sight, sound, smell, taste, and touch of a man lead a woman to defilement, intoxication, craving, impurity, affliction, attachment, existence, clinging, a man’s form (I’m not sure what this word means here, perhaps rebirth as a man?), and suffering of “going and coming” (death and rebirth) for a long time.
The Buddha is indeed pragmatic, but he is also wise and compassionate at the same time. There is no way that the Buddha would think of misogynistic rules as “effective solution to govern groups of disciples from danger”. Misogyny is clearly unwholesome. Those who make unwholesome deeds by body, speech, and mind are fools according to AN 3.6/EA 22.6; the Buddha is definitely not a fool.
The fact that misogynist passages are later additions to the early discourses is very clear. In AN 10.75 (its parallel is SA 990), the laywoman Migasala was criticised as “an incompetent matron, with a matron’s wit”; this doesn’t occur in SA 990 at all. In AN 7.63 (one of its parallels is EA 51.9), the laywoman Sujata was criticised as “what’s with the people making that dreadful racket in your home? You’d think it was fishermen hauling in a catch!”; this passage is entirely absent in EA 51.9. In MN 146 (its parallels are SA 276 and a discourse in the Mulasarvastivada Vinaya in both Chinese and Tibetan), the nuns are portrayed in a negative light and Venerable Nandaka acts confidently in not teaching the nuns; its the opposite in SA 276 and the Mulasarvastivada Vinaya in both Chinese and Tibetan according to Venerable Analayo’s Attitudes Towards Nuns – A Case Study of the Nandakovāda in the Light of its Parallels. In DN 16, there’s a passage where Venerable Ananda is told to avoid women; such passage is absent in its reconsctructed Sanskrit parallel according to Venerable Sujato in What the Buddha said to Ānanda about women: some textual issues. These are among some examples where misogynist passages are later additions to the early discourses
The monastic rules, as a whole, are later than the early discourses, so it is natural that there would be many later additions and alterations to them.
According to Venerable Analayo’s The Foundation History of the Nuns’ Order, he talked about a particular passage in some of the monastic rules regarding the Buddha’s refusal of ordaining women at first that:
The Mahīśāsaka, Mūlasarvāstivāda, and Sarvāstivāda versions report that he suggested an alternative of shaving off the hair and donning robes, apparently so as to cultivate a life of celibacy in a protected environment at home.
Misogynistic rules in the monastic rules have been studied in detail by Venerable Analayo and Venerable Sujato, and it seems that those rules are a result of redactions. They are either entirely later additions, or alterations of the existing texts. Venerable Analayo briefly talked about misogynist passages from 1:35:02 to 1:38:52 here: https://www.youtube.com/watch?v=YLT63llyQgI. In the topic of where Venerable Sujato talked about DN 16’s Sanskrit parallel where the passage about avoiding women is absent, he said the following:
More specifically, they agree with my theory that such passages, advocating a significantly harsher attitude to women, were the result of a “rigorist” movement within the Sangha around the time of the Second Council.
It’s true that the aim of practising the Buddha’s teachings is not “a utopia where all beings of all races, genders, ages, class, have equal civil rights”, but the Buddha does teach that one should give up bad conduct by way of body, speech, and mind; instead, one should develop good conduct by way of body, speech, and mind according to AN 2.11/SA 661/SA3 16.
In any case, we learn from the Buddha’s teachings that all things are nonself, and giving up what doesn’t belong to a self, especially unwholesomeness, will be for one’s welfare and happiness. So, according to the Buddha’s teachings towards both mendicants and laypeople, we should ultimately aim for extinguishment.