I just took a quick look at that page to see which leaders were quoted, what they said and when they said it. They're direct quotes, so obviously they were made. Still, I grew up in the Church (for 70 years, actually) and I really didn't find that kind of language all that pervasive. It cropped up from time to time, but I guess I just never focused on it. Fortunately, I was raised by parents who were kind of atypical. Had I even raised those issues to them (which I didn't), I have no doubt but that they would have told me never to feel pigeonholed into a life I wasn't comfortable with just because I was born female. I'll admit there were times when these extreme teachings bothered me. When my husband and I got married and deciding to postpone having children for over 9 years, I remember feeling judged by a lot of other women in the Church (never by my parents, though), and when I continued to have a career after my two kids were born, I also took a bit of flack. Interestingly, though, I felt the disapproval from my peers and not from anybody in a leadership position, certainly not from men.
As to why I stay... I still believe the core doctrines of the Church. When I think back over all the years I've spent in the Church, I realize that the doctrines I believe (i.e. the nature of God, God the Father's relationship to His Son, Jesus Christ, the idea of a pre-mortal life, a life where our best is expected of us but where we don't have to worry about being perfect, a post-mortal Spirit World where those who never heard the gospel on earth will still have a chance to accept it, a heaven which is ultimately large enough to be an eternal home for virtually everyone who has ever lived. No other church teaches the doctrines that have literally made me who I am today. These are eternal doctrines; they're not going to change over time. Policies are going to change continually, just as society changes continuously. Different things are going to be emphasized over the period of anyone's life, and the attitudes of people like me are going to slowly but surely influence the way the Church operates. As younger men are called to the highest leadership positions, that's going to help, too. But if we all bailed because of the things we didn't like, eventually the Church would cease to exist. The Church isn't just Boyd K. Packer or Spencer W. Kimball's church; it's also my church and their attitudes aren't going to push me away.
One final thought -- kind of to re-emphasize how I started my response. Maybe that Facebook page is called "Stuff You Missed in Sunday School" because the quotes found on the page were never the focus on the gospel. I might have heard one of those quotes on occasion (doubtless, I did), but they were more than offset by everything else I was taught. I learned to zero in on the 99% of what I was taught in Sunday School and just brush off the 1% that would have messed up my life had I taken it too seriously.