In our previous paper (paper I), we proposed a simple physical model that may universally describe glass-transition phenomena from the strong to fragile limit. It is based on the idea that in any liquid there always exist two competing orderings, which lead to two types of local structures frustrated with each other: (i) normal-liquid structures and (ii) locally favored structures. Here we demonstrate that this frustration, which causes an extra energy barrier for crystal nucleation, can be an additional physical factor to make vitrification easier. It can be regarded as impurity effects on crystallization. This idea provides us with a simple physical criterion for vitrification, which is consistent with the well-known empirical laws. We also check several main predictions of our model. According to our model, the melting temperature of the corresponding pure system free from disorder effects, T-m*, is a key temperature: Below it, a system starts to have special dynamic features peculiar to the Griffiths phase known in the field of random-spin systems, which is characterized by a complex free-energy landscape. We stress that this prediction is specific in the sense that T-m* is directly related to the real melting point T-m, which is an intrinsic physical property of the material. In our view, a stronger liquid suffers from stronger disorder effects due to a higher concentration of locally favored structures. This leads to a larger distance between T-m* and the Vogel-Fulcher temperature T-0 for a stronger liquid, which is consistent with experimental results. Finally, the effect of pressure on the fragility is discussed in the light of our two-order-parameter model of liquids.