Do most chemical reactions happen at constant pressure?
What happens during a reaction is important insofar as reversibility is concerned (a reversible path being a special way of performing a reaction in which equilibrium is sustained throughout), but not when computing $\Delta H$. The condition $\Delta H = q$ is derived from the basic law of conservation of energy and the definition of enthalpy. Because H is a state function, to compute the change in its value you need to know only the initial and final states of the system (and of course the values of H corresponding to those states). For $\Delta H = q$ to hold requires that (1) only expansion (pV) work be performed and that (2) the initial and final pressures of the system equal each other.
You can dream up scenarios in which you start at mechanical equilibrium (balanced pressure) between surroundings and system but end up at a constrained system volume (rigid). For instance, if an explosion moves a piston to a constrained setting within a cylinder. In that case it is not true that the enthalpy change and heat are equal. On the other hand, if the explosion is constrained such that the volume can expand until pressures balance (and external constraining pressure is constant), then enthalpy change and heat are equal.