Is pigment necessary to obtain white milk paint?
Yes, pigment is necessary. The milk solution creates the binder for the pigment, which is basically the mechanism to deliver the pigment to the support/surface, but it is then intended to mostly evaporate during the drying phase, leaving the pigment behind. What you're seeing now is, basically, an almost transparent darkening of the wood surface which is the remnants of the binder material.

You should be able to find dry pigment online. You're looking for, in this case, titanium white which is PW6 and it's generally available. I'm not going to give free advertising, but if you Google "titanium white dry pigment" then you'll find sources very quickly. :)