Geez,,,,,,Sounds like your contractor might be looking out for your best interest.That Drip is compared to gold and fine linens these days.With the money he is saving you,it might be possible for that cruise to italy you have always wanted to take.Or more like a couple of pork steaks a can of corn and a bottle of 30 dime wine.(Boones Farm Strawberry Hill)
Why waste time and effort to do a reroof and reuse anything,,,,,sorta takes the meaning of a new roof out of the picture.IMHO
When the neighbor comes by to yak,,,,,he can say hey nice roof.You can say yep all except the drip edge.
Its not saving you money as much as it is saving him time and effort.
Besides if you are buying the materials then he must be a labor only contractor.That is generally the "You get what you pay for" scenario.And no not all contractors who do work as labor only are knuckleheads.....Just the ones who convince or try to convince the customer to reuse something he is being payed to remove and replace.IMHO