The root causes of poverty and violent crime are people who have children but are not able or willing to properly educate them and support them financially, so the simplest fix is to prevent that from happening (by forcing them to have an abortion).
That said, an alternate solution is for the state to step in if parents can't or unconditionally, although some sort of population control might be required anyway.
Simple doesn't mean best or even acceptible. The simplest solution for stopping many forms of crime is to institute martial law, yet we do not do so because we view it as an extreme form of oppression and tyranny. I don't see any positive outcome from a government telling people under which circumstances they are allowed to procreate unless we are in dire straights (e.g. massive, world wide, irrecoverable famine. We're not there and we're not going to be there any time soon.)
Who pays for the abortion?
If someone conceives children without permission multiple times, wouldn't be it cheaper just to kill them so as to prevent them from consuming resources without permission? or should we merely forcibly sterilize them?
Yours is the stuff dystopian literature is made of. You want to severely limit freedom for the greater good, a tactic which never works.