Why would Roe have increased the number of out-of-wedlock births?
Because the possibility of abortion reduces the perceived cost of fornication (from $200k to raise a child to $250 to kill one). If the perceived cost of fornication declines, then fornication increases. More fornication=more out-of-wedlock births (OOWBs) since many mothers in practice chose not to commit infanticide and other government policies helped pay for the children.
The same effect was also produced by the existence of birth control. The mere existence of birth control technology (whether people use it or not) reduces the perceived cost of fornication.
Prior to 1960, the cost of OOWBs was high. Social disapprobation, lack of income to support mother and child, shotguns.
The 60s and 70s eliminated the disapprobation, shotguns, and lack of cash. More bastards resulted. If we happened to switch to calling bastards bastards, eliminated all public support (Medicaid), and brought out the (usually metaphorical) shotguns, OOWBs would plummet again.
I always thought that public employees (like Eugene) and recipients of government cash (SS) should be required as a condition of the receipt of such funds to verbally remonstrate fornicators, bastards, adulterers, and the unemployed. We know that government speech criticizing our dietary, health, and energy habits is common. What would be the problem to extending it to other human failings?
Instead we have a bastard likely to be nominated by a major party for president.