The basic theme here is not obvious to western Christians. The first generation of churches were expected to act like synagogues. A primary feature of synagogues was to implement and enforce the Hebrew customs for extended family households; the people in your synagogue and your family were the same people. For communal Hebrews this was instinctive and generally mandatory, but not so much with Gentiles, who likely joined a church without their extended family joining them. Both Jews and Gentiles were often cut off from family for confessing Christ. Your church may well include blood kin, but what mattered is that they were spiritual kin.
Paul advises Timothy to treat everyone like family, and to set the example in filial piety. When it’s necessary to pull someone up from a headlong rush into sin, do so in such a way as to make it clear you seek their best interest. This is nothing at all like the western habit of showing off how smart you are on some matter, and then pretending you are “just trying to help”. That would never fly in the Ancient Near East. This is not about you; give real help when it really helps.
A primary function of a Christian synagogue was carrying the burdens of a family. In a world where the majority of females got married at about age 15 to a man 10-20 years her senior, it was common for them to outlive their husbands. All the more was this true when men were exposed to more threats to life and limb than is common today. Meanwhile, the Old Testament bore a very strong statement insisting that extended families care for these widows.
Paul mentions “a window indeed” referring to someone who is cut off from any earthly family support. Is the church truly her only family? Then she should be supported and supportive. She should be wholly devoted to the church household in order for the church to feed and house her. If she has outside commitments, then she cannot give full attention to the church to justify such support.
If she’s below age 60, she’s too likely to get lonely for male companionship. The image here is someone still sexually active. Paul makes it clear he’s referring to putting these women on the church payroll as staff. These were something like modern nuns, though not so rigorously organized as is common today. Still, their activities were rather similar, emphasizing service and worship within the church community.
The concept of eldership was not so foreign, but Paul saw the need to advise Timothy on the uniquely Christian version of it. Again, this is family. Elders were effectively heads of households within the wider church community. Paul says they deserve support, as well, as the Greek word time refers to both honor and honorarium (“double honor”). It’s a privileged position; they should not be easy to take down. By the same token, any rebuke must be in the presence of those they lead, so that the other elders learn to take their roles seriously.
It is vitally necessary to grant some men a higher privilege; there’s no other way churches can exist. It’s rooted in the reality of dealing with humans. But by the same token, it must be obvious to everyone involved that there’s no human favoritism. It’s a privilege, not a right. Men are chosen for the role because God uses them that way even without the privileges. Fit elders would not argue with a godly rebuke.
Regarding the final verses here, I wrote elsewhere: “Paul… also advised him to drink wine, because water alone was not only regarded as extreme asceticism with most people, but wine would offset the effects of questionable water quality in that part of the world. Timothy need not be embarrassed how it would look. In the real world, we all know a sinner may not be obvious at first, but eventually his evil will slip out. The same with a good man, whose actions may at first seem a little odd. Once they saw Timothy acting truly magisterial, silly questions become obviously silly.”