If you want the patch to be seamless you need to remove the door, sand lightly the entire door. Apply a coat of the stain, wipe off--the worn areas should be the only place the stain "takes" then poly the entire door.

I have the same problem with some of my cherry cabinetsWere you able to sand/stain to fix it?

I did this same thing on my oak back in the day and it works well. "Oils" are a band aid and could get you through a party but they don't last very long. The more layers of poly you put on the longer it will last.