Why is Black used for interest rate options pricing instead of Black-Scholes? Why are we more interested in Future rates instead of Spot rates when it comes to interest rate options? Basically, why can't we treat interest rates like stocks when pricing swaptions etc.?
It's the forward rate which is fundamental to pricing for both stocks and interest rates. In the case of interest rates (unlike stocks) , it's difficult to compute the forward rate given the spot rate. Eg knowing the 10yr swap rate does not allow you to calculate the 1yr-10yr forward rate. The latter depends on the 11yr and 1yr parts of the curve for example. Hence in rates models the forward rate is used directly.
A point about modeling: in order to use the Brownian diffusion model, we need the underlying to be a martingale in some measure. For stocks , the stock price is s martingale in the money market measure. For interest rates, the forward rate is a martingale in the forward annuity measure (i.e. The value of a 1bp annuity for the forward period). So, modeling with the forward rate is 'nice'. As far as I know you cannot do that with the spot starting swap rate.