For all the reasons others have described, this is problematic. However, I propose a middle ground: develop permanent, reversible, side-effect-free birth control, and apply it to every child at 10 years old. When you turn 18, you can have it removed. You just need to show up at a government office, sign a form, and have the procedure completed. It is completely free, and you are out the door in an hour. The treatment can be reapplied at any time.

What happens? No more accidental pregnancies. No more getting knocked up in high school. No more scares after one night stands. No more becoming impregnated by a rapist. Everyone can fuck to their heart's content, but babies only get made if both people actually want a baby. Most of the problems you are talking about typically occur when either one or both of the parents don't want or weren't expecting a child. Make pregnancy opt-in, and you'll solve 90% of the problems.