In my area, anyway, lack of eave flashings is a invitation to water backup under the shingles from ice at the gutters and eaves.
Drip edge is "recommended" (not "required") by most shingle manufacturer's, but is not required by the IRC.
In your case the manufacturer's installation instructions say:
"NON-CORRODING METAL DRIP EDGES: Recommended along rake and eave edges on all decks, especially plywood decks."
and the installation instructions illustrate dip edges for both standard and low-slope applications.
So the roofer is technically correct when he says the drip edge is not "required", but it's also the case that the roof is not installed in accordance with the manufacturer's written recommendations - which IMO is a pretty low standard to shoot for.
BTW, are gutter end separations from adjacent walls:
hold-backs from roofs:
and kick out flashings:
present as/where required by the James Hardi installation instructions?