But that begs the question, “who is to blame for it being ‘innate’?” (If indeed it is, I’m not convinced it is)

Why a particular person has a particular cross is a usually a mystery beyond human understanding, but each person has his own cross to deal with. Jesus made that clear:
Then Jesus told his disciples, “If anyone would come after me, let him deny himself and take up his cross and follow me. . . Matthew 16:24

Crosses come in many forms. For example, there is evidence for a genetic component to alcoholism. People have that trait may need to abstain completely from alcohol, while others have no issues with moderate drinking. That may be a cross that they have to deal with.

As you quoted from John about Jesus’s explanation of the reason for the man to have been born blind:
. . . it was in order that the works of God might be displayed in him. John 9:3

Perhaps if there were no personal crosses, there would be no opportunity to display the works of God.