Why is it necessary to consider infinitesimal changes in p,V,T for H,U and G given that they're state functions?
It is important to make clear any requirement that a variable be constant for each equation you write.  

To go from

$$\mathrm{d}G = V\,\mathrm{d}p - S\,\mathrm{d}T$$

to

$$\mathrm{d}G = V\,\mathrm{d}p,$$ 

you would need to assume temperature is constant. 

And to go from

$$\mathrm{d}G = V\,\mathrm{d}p$$

to

$$\Delta G = V(p_2 - p_1)$$

you would need to assume volume is constant.  

If temperature and volume are both constant, the pressure is constant also.  

So your final equation is only saying: when temperature, pressure and volume are constant, there is no change in Gibbs free energy.