Yes, full coverage is needed. And not just of the lower lattitudes from the ecliptic plane. The original Solar Orbiter proposal plan A was for a highly inclined orbit passing over the poles of the sun. But this was too expensive and instead they went with just another generic spectroscopic imager in the ecliptic plane. It was such a disappointment.

I wonder if the 4m DKIST on Earth would have higher resolution photomosaic of the sun if it were used to do this one day? Probably. It's field of view is smaller but it can image features down to the high single km scale (~8km) on the photosphere.

The problem with this is that at 10km scale the features of the sun are changing far faster than at large scales. The rows of exposures' tops and bottoms would not match very well assuming a normal raster scan. The higher the resolution the smaller the timespan you have to take the full disk image. And the higher the resolution the smaller your FOV is. It's a rough situation.

Why? How would knowledge benefit, either directly or indirectly?

I sometimes wonder if we should have probes orbiting each planet and the sun, with the most generically useful sensors, for research that isn't planned decades in advance. But I really don't know if that would be a good investment of scarce scientific resources.