Partials can help the lockout, but I recommended floor presses and 2 boards because they strengthen that middle range which is where OP is weakest, true lockout work would be 3 and 4 boards, high pin presses, etc. In Jim's bench video, if I remember correctly, he shows both ways of benching. Typically you will see a shirted bencher benching in an arc patter. This is because, from what I have seen, they will bring the bar lower and flaring the elbows hard makes it easier to lockout the weight when the shirt stops helping.
Personally, my training partner can easily out deadlift me by 100lbs and out squat me by almost as much, but I'm out benching pressing him by almost 60lbs and I'm 20lbs lighter than he is and he has been training for much longer. We both have the same arm length and same grip on the bar. (I am a piss poor bencher. By that I mean I was deadlifting nearly 400lbs and could barely bench press 185lbs at 165 bodyweight. Now benching 295 at 175.) I attribute a lot of this to strengthening weak areas and have very good form. I press in a straight line, moderate arch, pinkies on the rings. He presses in an arc, big arch, pinkies on the rings. A lot of the time he will miss lighter weights simply because he flares his elbows, gets loose, and hits the bar against the posts.
Think about it this way. With a bigger arch your upper torso is closer to being inverted. So in a technical sense, it would be more efficient and easier to control bar if you pushed it in a straight line. If you bench in a arc, then a big arch is only going to amplify that arc, so the more arch you have, the less off a C shape you would want to bench in.
While this is all anecdotal, I hope I kind of explained my reasoning well enough.
Edit: Saw that you mentioned that you weren't talking about a big arc, but just a slight elbow flare. This is ok and should happen naturally, but the problem you run into is people that can't quite seem to understand this concept. You tell them to flare their elbows and they slam them all the way out like they are trying to shoulder bench. And like I stated before, the bigger the arch the person has, the more likely it is that even a slight over flaring with cause them to miss.
(I also want to re clarify to anyone reading. I'm talking strictly about raw benching.)