"But I don't want to have so many immigrants that we can't turn them all into Americans. I want immigrants who respect womens' rights, consider racists to be social pariahs, find corruption to be intolerable, embrace gays, etc"
If I understand you correctly, you are making the assumption that things like respecting women's rights etc. are primarily an American virtue and not as much with other cultures ? I would agree with you generally that America is probably the most open country even with its challenges but I am not sure if your point is relevant in the context of immigration. Here is why. Immigrants come to America for many reasons. Your dad's was not wanting to bribe for every little thing and probably many other factors like that. But in my experience, most immigrants are just looking for better opportunity and america provides that. It is hard to objectively quantify their virtues, beliefs etc.
So what I am getting at is that your point seems ok in theory but very difficult to implement in reality. If you want to restrict immigration, it certainly cannot be done based on perceived virtues. Plenty of immigrants assimilate in the local culture while plenty preserve their own at the same time. Nothing wrong with both as long you are not hurting anyone. In fact, what makes America special is you could look different, speak different, eat different but at the end of day, you are American as long as you have some common core beliefs including hard work, take nothing for granted, the drive to succeed and the perception of self independence.