this is new to me. I have seen steel doors that are primed and with a wood grain ready to paint, but not stain. Good idea.
My suggestion would be to get a good minwax product that does NOT have polyurethane mixed in. I have found that stain with poly is very hard to get many coats on and to darken the stain. Then after you achieve your darkness of the stain, apply a good UV protectant sealant to the door, just thinking out loud, don't know if UV would help since the door is steel skinned.
I would check this out further at a paint dealer store.