It looks like Biden avoided exposure to the draft by going to Law school. I believe at the time, going to law school allowed you to continue your 2S student draft deferment.
I remember seeing a lot of this. And I think that it is a component of how America ended up so over-lawyered. Furthermore, the lawyers we ended up over-lawyered by were the kind of person who became lawyers to avoid being in the military. Not a good combination.
You could, however, get a deferment if you were married and had a child, or even if you were married and your wife was expecting a child. It was not a sure thing. It varied from Draft Board to Draft Board. I suspect it depended on how much trouble your Draft Board had in meeting its quota. If Biden was married in 1966 and his wife became pregnant right away, that may have been how he avoided the draft. Biden grew up in or around New Castle, Delaware, and that was probably where he registered for the draft. That is a fairly blue collar area, and Draft Boards in such areas typically had little problem meeting their quotas.