If I’m understanding the concepts correctly:
A.) The value of the floating leg of an IRS is only the payment at the next settlement date + notional amount discounted back to today.
2.) The value of the fixed leg, however, is the present value of all of the fixed payments + the notional amount.
In solving for the fixed rate that would set the value of a swap at origination to zero, why do we equate A & B when A only uses a singular payment whereas B uses the aggregate?
At inception (t = 0), we set the PV of fixed leg equals to the PV of the floating leg.
PV of fixed leg is the PV of all fixed payments and notional amount.
PV of floating leg is the PV of all floating payments and notional amount. The PV of the floating leg at inception is equals to the swap notional.
For example, if you have a 1-year swap with semiannual payments, referenced to 6-month LIBOR, and given that swap notional = $100 and the spot rate and forward rates:
6-month LIBOR today = 3%
6-month LIBOR 6 months from today = 5%
For the floating leg, the floating payments 1 year later (maturity) is $100 x 5%/2 = $2.5. The PV of the payment in 1 year back to Month 6 is:
($100 + $2.5)/(1 + 0.05/2) = $100
The floating payment at Month 6 is $100 x 3%/2 = $1.50. So, the PV of the cash flows at Month 6 back to t = 0 is:
($100 + $1.50)/(1 + 0.03/2) = $100 <— (PV of floating leg at t = 0)
The assumption is that the floating rate will reset to the market rate, and that the market rate is the appropriate discount rate for the swap payments. Given those assumptions, the value of the floating-rate bond used in the swap equation will reset to par at the next coupon date. That (par) value is the present value of all future coupon plus principal payments, so the value of the floating leg does, in fact, include the aggregate of all payments; it simply does so sneakily.
@S2000magician Thanks so much for the response magician! I went through some mental gymnastics and I can see how par value actually captures the present value of all future coupon + principal payments!
Thank you Fino! Your post along with magician’s gave me the much needed push to finally grasp the concept.
Just to double-check my intuition - If we are to extend your hypothetical scenario to 18 months with 6-month LIBOR 12 months from today = 7%:
PV of payment in 1.5 years back to Month 12 is:
($100 + $3.5)/(1 + .07/2) = $100
Since the floating payment at Month 12 is $2.5, we now have:
( $100 + $2.5)/(1 + .05/2) = $100
So on and so forth.
P.S. Sorry all if the thread looks messy, still figuring out how to efficiently post/reference
Spot on 
Anyway, we have to be careful with all these payments methods!
What that even means !!