Also, so far as I know, no-one ever explicitly derives the relations of the Yangian from the conditions that it be a Hopf algebra quantization of the Lie bialgebra, but Drinfeld explains it in his article called Quantum Groups (Proc. ICM Berkeley, 1986). If you assume the coproduct takes a certain form on the Lie algebra (the most obvious choice given that it is a quantization of that Lie bialgebra) then impose the condition that the Hopf algebra coproduct be an algebra homomorphism then you can derive the relations of the Yangian in the first presentation term by term, using the fact that it is a homogeneous (graded) deformation.
Personally I tried to do this but I found I had to assume little things along the way, like that the RHS of equation (13) in that paper is a symmetric sum of the orthonormal basis elements of the Lie algebra. I couldn't really understand why I had to do this, but I guess it's probably obvious to people who are smarter than me. At the end you end up with an algebra that you can then prove is isomorphic to the second presentation he gives, and then get a PBW theorem based on this presentation and verify that it is a quantization of the Lie bialgebra after all (I still working on understanding this part). Then given the uniqueness, which is meant to follow from cohomological arguments described in Section 9 of that same Quantum Groups paper, you don't need to justify any assumptions you make along the way. It seems that understanding the cohomology is key.