I believe that child-rearing should be the responsibility of both parents and that, whilst the roles within that partnership may be different, they are nevertheless equal in importance. In some societies, it has been made easier over the years for single parents to raise children on their own. However, this does not mean that the traditional family, with both parents providing emotional support and role-models for their children, is not the most satisfactory way of bringing up children.
Of crucial importance, in my opinion, is how we define 'responsible for bringing the children up'. At its simplest, it could mean giving the financial support necessary to provide a home, food and clothes and making sure the child is safe and receives an adequate education. This would be the basic definition.
There is, however, another possible way of defining that part of the quotation. That would say it is not just the fathers responsibility to provide the basics for his children, ; while his wife involves herself in the everyday activity of bringing them up. Rather, he should share those daily duties, spend as much time as his job allows with his children, play with them, read to them, help directly with their education, participate very fully in their lives and encourage them to share his.
It is this second, fuller, concept of 'fatherhood' that I am in favour of, although I also realise how difficult it is to achieve sometimes. The economic and employment situation in many countries means that jobs are getting more, not less, stressful, requiring long hours and perhaps long journeys to work as well. Therefore it may remain for many a desirable ideal rather than an achievable reality.