We are carnal (flesh). We are sold under sin.When Paul was speaking of the sin nature, he said in Rom. 7:14,
"...but I am carnal, sold under sin."
Sin is sure with the sin nature. Paul taught us in Rom 6-8 that we can control that sin nature by walking in the Spirit.
So the sin nature plays a huge roll in our need for a Savior.
But we are talking about a "sin nature". This is a category foreign to Scripture. Sin occurs not when we desire something, not when that desire becomes temptation, but when we give into that desire. Man sets his eyes on the flesh (on the carnal) and sins (and is sold under sin).
This does not mean that Jesus did not have a human nature like us. He set His eyes on the Spirit. He overcame.
Think of it this way - If Jesus was tempted in all points like us then He had to have been tempted by the flesh (the carnal). Would it be sin for Jesus to go into the wilderness to fast and pray and yet eat because He was hungry? Yes.
It is not an issue of ability but of the will - of Jesus desiring to do the will of the Father.