I don't believe child killing would require prior convictions in order to get the death penalty in Texas.
https://manshoorylaw.com/blog/difference-between-1st-2nd-and...
It seems they have portrayed him as a cold, calculating murderer.
Second degree murder implies at minimum extreme indifference to human life if not intent to harm whereas he took her to the doctor then the hospital and gave her medicine then deemed medically appropriate and only now known to be harmful.
Where are you seeing the extreme indifference to human life?
Noting here that using 'child killing' in this context loads the term. It unhelpfully stirs intended and unintended deaths together and conflates murder with awful mistakes made by poorly equipped, poorly informed or overpressured parents.
Parents who've lost infants after rolling onto them while sleeping - they ought not be lumped together with people who've killed a child thru a planned, horrific act.