Why is phase of current assumed to be 0 when deriving formula for instantaneous power in an AC circuit
The instantaneous power is the product of instantaneous voltage and current. These two can be out of phase. Current can be lagging or leading the voltage, but you can also turn that voltage is leading or lagging the current, it doesn't make any difference in the math. Now one of them is used as reference and the other has a phase shift with respect to reference (the other).
Why would you need two phase differences? Which is now reference, universal time? Just keep it simple.
Your derivation of formula seems wrong to me, if you do the math carefully, then you should get the same result.
EDIT:

$$sin(\omega t+\alpha)\cdot sin(\omega t +\beta)=\dfrac{1}{2}\cdot ( cos(\alpha - \beta) - cos(2\omega t + \alpha + \beta))$$
So, your calc is good and their is wrong.