I'm glad they didn't give us that ending. It sounds even dumber than what we ended up with. 

Short answer: The show ended as a drama bathed in blood and death. Reverting it to a romcom and then trying to make sense of the time travel portion doesn't work. It's like turning in a baking soda volcano for your senior science project in college.  Long Answer: How would they discover that he was a gourmet after a car accident? Did they run into a guy with their car and then feed him right away? (Never do this, btw) Has the Korean language changed so little in 500 years that they could understand him? His leg had already been injured during the fight with his uncle, and he was covered in blood. Did they take him to a hospital? How did he explain his lack of ID, how did he adapt to all the weirdness (even 21st century people get scared in hospitals) ? They love to stick IVs in you, how did he accept that?  Did he mention to no one who he was? Feign amnesia? If so, there would have been news reports, right? Was he magically cured by crossing time? Then why did she still have the horrible cut? Plus, look at how the time travel worked for her,  she appeared in the past with a body, that body disintegrated in the past to have her soul drop into the body in the present which was already in a hospital bed. So for most of the show she hadtwo bodies, one in the present one in the past. But one soul, it spent time in the past, came back to the present.   But his body came with him, disappearing in the past to appear in the present. So he has just the one body. It makes no sense.