Is it okay to face nail the drip edge to the fascia?
TBH, it would probably be pretty difficult to lift the edge of the shingles enough to actually be able to nail the drip edge down from above. The shingles should be nailed pretty close to the fascia to prevent them from lifting in the wind, so (IMHO) you're doing pretty good that they were able to slip the drip edge under there to begin with.
Personally, though, I agree with your concern and would probably put a little smear of clear or white silicone over each of those nails to ensure water doesn't get through. Whether it's strictly necessary to do so, I couldn't say, but that's what I'd do.