The possible solution to this problem is that perhaps the rules were different when the Galaxy was young. Perhaps star formation in the early epochs of the universe strongly favored higher mass stars, the stars that would eventually make the population of white dwarfs in the halo. At the same time it would have formed relatively few low-mass stars. There are thousands of white dwarfs known in the disk and the study of those has been a major sub-field in astronomy since white dwarfs were concieved and the first example found orbiting the star Sirius. Stars in the disk that evolve into white dwarfs will remain in the disk.
Spectroscopy is the astronomer's most powerful tool.
It is a technique which allows one to determine the brighness of an object as a fuction of wavelength of light, or equivalently color. It turns out that the properties of atoms and molecules leave tell-tale signs in the spectra of virtually all things. By measuring the spectrum of an object one can determine what sorts of elements and molecules exist in the object. In the case of stars the spectrum reveals the contents of only the atmosphere. The interior of the star is shrouded by the atmosphere and cannot be observed directly using spectroscopy.