You are correct. Sure it looks messy and bloody but scientifically speaking let us compare between beheading and hanging.
One you have beheading the sword will hit the medulla oblongata which will automatically cut off all pain nerve impulses the pain that will ne felt is for like quarter of a second but then that is it.
Hanging you have two options. One is the axis will hit the medulla cutting off the cardiac and respiratory centers which mean the one who is hanged will still feel pain for am few seconds before death in the mean time he will experience tue pain of both a cardiac arrest and pulmonary thrombosis. The other is suffocation until death. Sure it is not bloody or messy but you decide which is more humane.