Why don't we let ourselves follow our child's lead

When we become parents, it seems that we automatically have to build into our parenting scripts that we need to worry constantly and overprotect our child(ren) from everything. Why is it that we do that? Is it society that adds that pressure through what we see in the media or from how our own families raised us? Or is it just that we look at the tiny little being in our arms and don't trust that he/she will be able to know