How is heat physically released in an exothermic process?
I think that you are on the right track. I try to explain how I see things below.

As a bond forms the reactants must absorb some energy from their surroundings so that the activation barrier between reactants and products may be surmounted. (By the Boltzmann distribution there are always a few collisions with far more energy than the average). At the top of the barrier is the transition state (TS), for this to be crossed the excess energy has to be taken away from the TS more rapidly that it may cross back to reactants. 

The transition state is effectively a very hot species; it has far more energy than the surrounding medium. This can be thought of as a species having a large number of vibrational and rotational quanta excited. As far as is known$^*$ a transition state lasts for only a few tens of femtoseconds to a few picoseconds because vibrational/rotational energy is lost very rapidly to the surroundings. This excess energy is carried away also as kinetic energy of the products if, say, two molecules of product are formed. As time progresses this energy excites more and more vibrational and rotational modes in the surrounding medium, which may be a solvent or protein's backbone and residues. Eventually this energy diffuses away from the reaction site (via anharmonic vibrational couplings) and the product eventually has vibrational and rotational modes excited with quanta in accord with the prevailing temperature.

$*$ Although transition states are universal there is a paucity of actual, i.e. time resolved spectroscopic data on their properties.

(A more detailed approach has been given by Kramers (see M. Daune ' Molecular Biophysics for a discussion). In the condensed phase 'friction' or 'drag' on the reaction coordinate slows the reaction (compare to gas phase) and the reactants 'diffuse' towards and across the transition state. 'Friction' here means vibrational and rotational energy affecting the reaction coordinate, by impeding or promoting, by turns, the progress of the reaction. The Kramers approach allows a consistent approach to  gas phase reactions from isolated molecules, then in the presence various pressures of inert gas right through to reactions in viscous solution.)