Pressure exerted by a particle of ideal gas
Here's your confusion:
You need to consider two different things:

The momentum transfer per particle per collision.  There, since we assume an instantaneous collision, it doesn't make sense to try to figure out force from acceleration. [I suppose you could do this using limits, and maybe there are applications in which that does make sense, but adding that complication is completely unnecessary for introductory kinetic theory, given that there is a simple and direct way to calculate the effect of each collision.]  And the simple and direct way is to use the change in momentum, which is $2mv_x$.

Note: Your statement that &quot;The particle is assumed to travel perpendicularly towards one of the sides before bouncing back off it&quot; is incorrect.  Rather, in determining the momentum transfer per collision, we use the component of the velocity perpendicular to the wall, namely $v_x$.

The number of collisions per unit time.  Here is where the average time it takes for the particle to travel from collision to collision is relevant. This is where ${\Delta t}$ is being used -- not to calculate the momentum transfer/collision, but to determine the frequency of collisions per particle.  And you need the average momentum transfer/collision, and the frequency of collisions per particle, to determine the average force per unit area contributed per particle.

So, in summary: For momentum transfer per collision, we don't consider time, since we can get this directly from $2mv_x$.  For number of collisions per unit time, we need to account for the time between collisions.