Why are there so many dwarf stars (red and white) among the nearest stars, but none among the brightest stars?
Mainly because of temperatures and Sizes.
There is a different story for each type of dwarf star that we cannot see.
if you are considering Proxima-Centauri, Proxima-Centauri though is the closest Star to the Sun but at the Same time it is very much faint because of it's size and mainly because of it's temperature.
There is a Simple relationship between Luminosity of an Object vs it's area and temperature. It goes like this.
Proxima-Centauri is a Red-Dwarf, Red color indicates that it's temperature is below 5000
White Dwarf on the other hand are extremely hot, much more hotter than our own Sun in it's main sequence stage. This immense temperature of a white dwarf is mainly due to the pressure in the core. White Dwarfs are pretty faint and temperature is not the culprit this time. It's the Area of the White Dwarf that makes it pretty faint. The area of a typical White Dwarf if about the same as the Earth's Size so, it is very much difficult to spot such faint object at such distances considering even the nearest to us Sirius B at 8.6 Light Years.