Very, very few people would want a government agency deciding whether they could have sex or not.
Consider, also, that when consenting teenagers and young adults start exploring their sexuality, there's a great deal they don't know about mature relationships - that is part of exploration and maturing. One hopes they've been adequately advised on reproductive health, pregnancy, disease, and consent, alternate sexuality, etc., but face it, they are not gonna be mature for a while and they are gonna have sex, unlicensed or not.
Lawtonfogle, have you read The Handmaid's Tale, by Margaret Atwood? It is about a dystopian future in which, among other themes, sexuality is highly controlled as a response to a loss of fertility in the general population. Only certain people are permitted to have children, and every woman who can bear children must be controlled by the people with the permits - the government, IOW.
(I should mention, the book also portrays a society in which Christian fundamentalists govern and run society, you may find it insulting or offputting on that level).