It turns out that Under Defeat was already tailor-made for this sort of experimentation. Unlike other shmups, Under Defeat lets you angle your shots instead of always shooting straight up. Your angle is tied to your direction of movement. Move left, and the nose of your helicopter angles left, for example. Then, it stays facing that direction for as long as you hold the fire button.
Most of the work in making this game suitable for a wider display just involves making your ship turn more when you move, in order to cover more horizontal space. Simple! I believe the enemy spawn patterns are changed too, to emphasize stuff coming from the sides and have less stuff rush at you from the (now shorter) top of the screen, but ... to be honest, I'm pretty bad at Under Defeat, and generally too panicked to get good solid intel about that sort of thing.
What I do notice is that this game looks really nice when you can actually see it! In arcade mode, with everything zoomed out, it looks pretty good, but the game looks much better when you can see all the polygonal tanks, planes, and tiny (unharmed) horses underneath you close up. This is another advantage specific to Under Defeat, as it just happens to be polygonal and therefore more easily zoomable than a hand-drawn shmup.
It would take a great deal more work for other shooters to adapt to HDTVs in the way G.Rev has adapted Under Defeat, but it works in this case. And how often do you hear of something demonstrably new in this genre?