My grandmother was born out of wedlock and given up for adoption — but only after her 18-y.o. unwed mother tried desperately to raise her for the first 4 months of her life. Her biological mother wanted her so much that she actually tried to gain custody, via the court system, a few years later when marriage improved her economic circumstances... but my grandmother was already happily adopted by a family who adored her.

So yes, you're right — most women are FIERCELY protective of the babies they birth. No matter how unready or unable they were to keep those babies, or how unwanted the babies might have been before they were born.