1 Tim 3:1-7, can girls be bishops?

I got an email recently from a friend concerned about the issue of how complementarian and egalitarian readings of this passage in 1 Timothy 3:1-7 about the qualifications for overseers (KJV: bishops) seemed to come to totally different conclusions about the Greek, surely someone was right or wrong here?

image credit: https://www.southcoasttoday.com/story/news/nation-world/2006/11/05/female-bishop-becomes-head-episcopal/50406091007/

Now, disclaimers: 

1. If you weren't already aware, I should disclose that I am theologically egalitarian myself - although I hope I can come to grammatical issues without bias.
2. I am dubious as to the extent these debates can be solved via Greek grammar. Accurate reading of the Greek is very important (don't get me wrong), but even the hardest biblicist (if they are honest/self-aware) is also drawing on other resources to decide any controversial issue.  
3. Also I have only given this only a little time today, if you know better than me I'll be glad to be corrected. This is not an expert opinion but my best stab at the problem based on what I think I know.

The issue is that egalitarians (those who consider the Bible to allow women equal ability to lead,teach, etc, in the church and home) have argued that this passage contains no male pronouns and should be interpreted inclusively. This is because in Greek, while there is a neuter declension for nouns, when referring to a mixed group of people it will use the masculine plural. Thus, there is almost always some ambiguity in the Greek as to whether a plural masculine refers to only men or men and women, unless the author, context or narrative specifies - e.g. Mark 6:44 where Mark says there were 5000 who ate (masculine plural) but then specifies "men" (anēr). (NB. Hebrew is worse, it has no neuter at all, only male and female grammatical gender)

So, in many English translation of 1 Tim 3:1-7 (e.g. the NIV, below) we see the words "he" and "his" we would be forgiven for thinking that the text specified men, however, it is entirely possible to translate this section accurately without such pronouns, because the Greek does not use their equivalents here.

3 Here is a trustworthy saying: Whoever aspires to be an overseer desires a noble task. 2 Now the overseer is to be above reproach, faithful to his wife, temperate, self-controlled, respectable, hospitable, able to teach, 3 not given to drunkenness, not violent but gentle, not quarrelsome, not a lover of money. 4 He must manage his own family well and see that his children obey him, and he must do so in a manner worthy of full respect. 5 (If anyone does not know how to manage his own family, how can he take care of God’s church?) 6 He must not be a recent convert, or he may become conceited and fall under the same judgment as the devil. 7 He must also have a good reputation with outsiders, so that he will not fall into disgrace and into the devil’s trap.

In fairness to the NIV, the pronouns have probably been added to make the English easier to read, but have they made the passage exclusive to men when the original could be read more inclusively? Despite the lack of pronouns we do have being head of a household and being a one-woman man as qualifications here - could these include women also?

The people of the historical-cultural context of the letter would assume that most households are presided over by men, and only in exceptional cases would an independent women be head of a household. This is important for how the passage would be received, but in my view it is unfounded speculation to say whether Paul intended to include or exclude women here. Here is what I mean about Greek only getting us so far, the egal/comp debate was not on Paul's radar at all (it's a new thing) - and so our question is underdetermined by the text. It should go without saying, but Paul was not a complementarian or an egalitarian, but a 1st century Jewish follower of Jesus!

I think the real issue for an egalitarian reading of this passage is the lovely expression "one-woman man" (often horribly translated as "man of one wife/married only once") - but I think this, in historical context, is more likely to be about monogamy than divorce. That expression uses the specific Greek word "anēr" which exclusively refers to men as opposed to women. That said, given female heads of households would have most likely been single/widowed (think of Lydia, e.g.), a direction about monogamy (against polygamy and, more likely, sexual use of others in the household) would not exclude the celibate from ministry.

So, both sides of the comp/egal debate could argue this passage aligns with their view, but I don't think it really sides with either. Purely on the basis Greek grammar you could argue this passage doesn't exclude women but it hardly includes them either. On historical-contextual ground you could argue it would assume women were excluded - but neither view is explicitly asserted because (I think) the passage is concerned with the issue of character, not gender roles. 

Let me know what you think. :-)