I think EIE is a good working version, but I'm not convinced myself, and I don't feel I know enough about his life yet.

It's definitely not strange for EIEs to dominate in artistic fields or in fields like advertising, publicity, or public life in general.

If so, it blows my stereotype of EIEs as people who are highly social and successful (especially in management) but more inclined to work through others by shmoozing and telling people their "grand vision" and not so inclined to work out the details themselves.
I think your stereotype is largely correct for EIE leaders and managers. But artists and performers must master all aspects of their skill without delegating to others, so it doesn't quite apply. Composition is hardly a team endeavor.

In contrast to IEIs and ILIs in the arts, EIEs are usually noticeable, loud, and somewhat controversial. If, on top of that, they have actually mastered their art, it's easy for them to dominate the public's attention.

Beethoven doesn't seem SEI to me, either. Part of it is that the theme of struggle and overcoming difficulty is not very relevant to the enjoyment-oriented SEI.

But I wonder who would represent the point of view the best (?).
Good question. I don't know any off the top of my head, and types in general seem to gravitate more to the visual arts than to music composition. I would look for composers with a "full," melodic sound without excess musical and structural complexity. A possibility that comes to mind is Grieg. Maybe you can think of others.