Why doesn't pH = pKa1 in the buffer zone for this titration?
Your acid dissociation constants are $10^{-0.6}$ and $10^{-1.74}$ whereas your thiosulfate species have only a total molar concentration of $10^{-2.00}$. When the dissociation constants of an acidic solute are greater than the solute concentration the solute is rendered like a strong acid by dilution.  If you have 0.01 M of an acid with a $pK_a$ of 1.74, and you use the equilibrium relation to determine how much of the acid is dissociated, you find that in fact most of the acid is dissociated.

The proposed use of the H-H equation then does not work because this equation assumes that aqueous hydrogen or hydroxide ions are only a small part of the total solute concentration.  When the solute becomes like a strong acid (or strong base) as above, it floods the solution with aqueous hydrogen or hydroxide ions and the assumption needed for the H-H equation fails.