Iodine formation reaction
A bit more complex per a source 'Kinetics and mechanism of the iodate-iodide reaction and other related reactions', to quote from an article in the journal Physical Chemistry Chemical Physics (November 1999):

The reduction of iodate by iodide, known as the Dushman reaction, is one of the components of the oscillating Bray-Liebhafsky reaction. The large body of work devoted to the kinetics of the Dushman reaction gives seemingly contradictory results. This work analyses and summarizes them and shows that they are special cases of a complicated kinetics and that they can be explained by a common mechanism. This mechanism can be generalized to all the XO3- + Y- reactions (X, Y = I, Br or Cl). They all involve an asymmetric intermediate YXO2.

See reaction system depicted in Equations (8) to (13). Here is a partial discussion of the system from the available PDF to quote:

The essential hypothesis of this mechanism is the existence of as a definite intermediate species and not as I2O2 an activated complex. Its existence was already proposed in  1930 by Bray [17] in order to explain the classical rate law. We make no distinction between I2O2 and H2I2O3 (I2O2 + H2O) as these two forms are kinetically equivalent. In reaction (12), B~ represents the anion of the buffer....

As to the question on the role of H+, note that the starting Equations (8) and (9) of the system both involve H+, which results in the cited intermediate $\ce{I2O2}$:

$\ce{IO3- + H+ ⇌ IO3H}$  (8)
$\ce{IO3H + I- + H+ ⇌ I2O2 + H2O}$  (9)