But a woman can choose to not have the child.
It is only fair to give the man the same right.
This is not right. You're right in that many cases a woman can choose to not have the child with or without consent of the father, but she is then faced with the real consequences of having to have an abortion, whether emotional, physical, or other. Every woman will deal with an abortion differently, and I'm not arguing that all women feel consequences, but certainly there is a burden there that must be dealt with. A man would not have to deal with these consequences - he could just literally walk away. To me, this is a significant moral hazard.
There's just not a subtle way to put except that when a man has sex with a woman there is always the possibility of pregnancy. Condoms break, the pill is not 100% effective, vasectomies sometimes fail.
That's not to say there aren't problems with how paternity is treated in our current legal system. I just don't think downplaying the responsibilities men face is a good strategy here.