The difference between edge and face select could also be that in face select, edges are dimmed. As such, the element type that corresponds to the selection mode would be the most prominent.
This solution doesn’t require edges to be thick to differentiate edge select from face select modes.
As for face dots in general, they are a useful mesh debugging tool, just like normal display or face orientation, which you can enable when needed. But they do create a lot of clutter, which is especially problematic with dense meshes, where the face dots can actually make topology unreadable.
What we should do, is keep face dots off by default, but obviously keep them as an overlay option. Secondarily, we should explore other ways to differentiate edge select and face select modes, without resorting to the thick edges.