What to use between tiles and door?
... is that just over the top and too visually intrusive?


Yes, but that would be opinion; beauty is in the eye of the beerholder


  Why not silicone?


Caulking would be better here (not necessarily silicone) because of vibration induced cracking of hard grout (as seen in your picture). The problem with caulk is color matching, that's why I would use paintable caulk and carry the door jamb paint over the caulk and up to the tile line.