Having the chevron to the side of the avatar is arguably unnecessary at
this point; most apps out there (Reddit, YouTube, etc.), including here
on GitHub, have the avatar without any icon, pointing to how there
doesn't seem to be a problem with knowing that there's a menu behind it.
Therefore, figured we could simplify the UI a bit more here.

This also looks better on platforms where the window controls are on the
right (Linux/Windows).

Release Notes:

- N/A