the gutter that is there now and the gutter that was on the left didnt do anything for your situation.
The most simple way is to remove the overhang completely.
And then install 6 inch seamless gutter system with a splash gaurd at the valley.
That would be the cheapest way. but then you would need upgraded door, door sill,door trim.
because even though you have completely solved water run off problem...doors need overhangs just for windy storms.
So ya, a flat roof with small pitch would be best so you can still have a complete gutter system.
You sound like you know what to do.
I would find a state certified roofing contractor with a building liscense also.
But one that was a roofer before he was a builder.
Usually, and mostly, its the other way around, and those suck.
Usually the ones that are builders who add on roofing,
Have never done any roofing.