As far as I can see, Buddha taught extensively to lay people how to
treat each other friendly & compassionately, but omitted the bodily
intimacy aspect of it.
Correct.
It's largely sensual desire that binds human couples together,
I think the Buddha would disagree with the above. In AN 4.55 and AN 4.53, the Buddha says it is having the same faith (goals), the same morality, the same sacrifice/generosity and the same wisdom that binds human couples together.
Since the 1960s sexual revolution, the amount of sex has enormously increased and the amount of "binding of couples" has enormously decreased. This appears to show that it is not largely sensual desire that binds human couples together.
Historically, it was obviously family, children & religious norms that bound couples together.
Unchastity is the taint in a woman
"Unchastity" is merely a translation of a word (duccarita), which most generally simply means "misconduct". The English word "chastity" can mean "celibacy" (which in Pali is "abrahmacariyā") but it can also mean "the state or practice of refraining from extramarital sexual intercourse", which is what Dhp 242 is obviously referring to.
AN 6.52 says: "Itthī puttādhiṭṭhānā: Women have sons as their support/mainstay". In other words, women should not use sexual intercourse as their support/mainstay or social refuge.
Q2: Could it be that a majority of Buddha's teaching are timeless, but
few are open for change, due to different societal context etc.?
(e.g., 3rd precept)
No. It is a timeless religious teaching the people should not have extramarital sexual intercourse. Anyone that believes extramarital sexual intercourse is not unwholesome & not harmful and a life-style possibly will not attain the Noble Path. MN 115 refers to what is possible and impossible as says:
They understand: ‘It’s impossible that someone who has engaged in bad
conduct of body, speech, and mind, could for that reason alone, when
their body breaks up, after death, be reborn in a good place, a
heavenly realm.
MN 115
The Buddha taught (DN 31) parents have a duty to arrange the marriage of their children. Therefore, in the 3rd precept, the word "protection" means this, namely, children and particularly a daughter is "protected" by her family until she is married. The Buddhist Commentaries are also very clear about this (which the monk named Yuttadhammo posted somewhere on this forum).
Q3: In the above quote, isn't it a bit biased to state that it's ONLY
a woman's taint? Why not a man? Why focus overly on women? Back in
those days, there were probably fixed gender roles, but I believe
society should progress & should rather acknowledge individuality even
among the genders. It seems that the Buddha is over-generalizing, or
did the miss the boat?
No. It is not biased at all. The question simply shows a misunderstanding of women and being beguiled/fooled by the modern social engineering or sexual exploitation of women. Women are different to men. Women have wombs and biological & psychological mechanisms created for both child bearing & controlling of families (nest building). There have been many scientific studies done on how sexual promiscuity affects female "bonding" instincts. Unlike men, in my experience, once women veer down the perilous path of sexual immorality or sexual promiscuity it is difficult for them to return to a state of wholesomeness. Women can develop "sexual addiction" in a manner which is a quest to over rejection & regain power. There are many studies & theories on this, such as merely this example: Female Sex Addicts: Using Sex for Power and Control. Even when such women eventually get married & have children, they will generally maintain the wrong views about sex, which they will impart to their children, who are then likely to commit sexual misconduct. Thus relationships in society, which we witness today in the West, often fall apart.
One should be aware, for example, it is reported 25% of Western women, over the age of 40, are using some type of antidepressant. Women are simply not designed by nature for sexual promiscuity because, as the Buddha taught (in AN 6.52), women have the strong tendency towards "domineeringness" and thus, in my experience, they generally struggle to admit wrong doing once they commit wrong doing. If one has any experience with women, one will find they get very angry, very upset & even very shattered when they are accused of serious wrong doing. If one examines Radical Feminism, for example, it will be found Radical Feminism rarely blames women for their problems and generally exclusively blames men for female problems. Any competent psychologist can explain this, such as this video called One Clinical Mistake That Can Heighten a Client’s Shame.
Women are nearly twice as likely as men to be taking the medications, the report found, with antidepressants used by 16.5 percent of females compared to just under 9 percent of males.
WebMD
Over the past decade, people have increasingly treated depression with medication: Starting in 1994, the number of antidepressant prescriptions written by doctors went up 400% over a 10-year period. And today, about 15% of women take an antidepressant. Among women age 40 to 59, that number is nearly 23%, according to the U.S. Centers for Disease Control and Prevention (CDC).
Woman's Day: Depression is common among American women, and antidepressant use is on the rise. Yet women tend to keep both a secret. Why aren't we discussing this more?
In summary, per MN 115, anyone that believes extramarital non-committed sexual intercourse is a lifestyle possibility will not attain the Noble Path. This is because they will not be free of the hindrance of sexual desire and have insufficient loving-kindness & compassion to empower their practice.