You're not wrong here. If the Lover only wants to provide sexual pleasure, he can't realistically expect the woman to stay around forever (since she requires a Provider). Most women also want children, so if the Lover does not want a child, it should be expected that she'll eventually move on to find a man that does want one.
This is still separate from the situation that is being presented in the above article: should a Provider provide for another Lover's child? If I was the Provider in this situation, my answer would be an adamant "no" because the only way this would have happened is if the woman did this discreetly without my knowledge since I absolutely would not have allowed it. However, the real answer is: it depends on the Provider-in-question's perspective. If he sees no problem in raising another man's genetic child, then fair play to him. Ultimately, the burden falls on him since the woman deceived him and he is not bound legally to raise a child that is not biologically his, so if he wants to place that burden on himself and continue raising that child, then he is more than welcome to do so.
In my view, the woman is taking an enormous risk by impregnating herself with a Lover's child and then secretly expecting a different Provider to take care of it. It is deceitful (assuming she purposely hid it, which is usually the case), and the Provider is not legally bound to financially provide for the child. In a "disaster" scenario, she would be relying on the courts to force her Lover to pay for child support after her Provider leaves and refuses to support her and the child. This is assuming the Lover even has the money to provide for the child at all.
Ultimately, the act of attempting to secretly get a man who does not want to raise another man's child to actually raise another man's child is a self-serving one for women, and it bears its own risks. If the man finds out, then he ultimately possesses the right to completely abandon that woman and the child if he feels that is better self-serving to him in the long run. He is not bound by the law, and he is not bound by any social repercussions if he chooses not to be.
All is fair in love and war. :)