[carpdad]

You will not see much of finish facia when gutter goes up. I would remove that 1x facia, add sheetmetal cover slipped under the drip edge, and add gutter. That 1x isn't going to last anyway. I think I see rake drip edge on top of wood rake? Z flashing could be slipped under the that drip edge to cover the rake trim.
A lesson for you. No communication with roofers and roofers who don't think. In nj, and it's my experience that any unsupervised trade will cut corners, small and big. Even better if they can screw you, rotorooter specifically. I paid full homedepot price, only about 500 square feet roof, simple gable, and there were few problems that needed short discussions with complaining faces.

 

[carpdad]

Drip edge is made so it kicks bits of rain away from facia into gutter. Without the kick out edge, water will follow the face of the roofing and facia and it can go to early rots. But if your roof is not low slope and shingles are overhanging, less chance the surface tension of water will follow that route.