I understand that this is a process that is very fast compared to everyday timescales, but as far as I can see, the reason it solves the horizon problem, if it does, is because it makes the early expansion of the universe extremely slow compared to what one might otherwise expect, and what was in fact thought.
Well yes. In simple terms, The inflation model creates more conformal time due to the huge increase in the scale factor. So with the help of inflation, there is "enough time" to create a heat equilibrium between two antipodal points. Without inflation, there's "not enough time" to create this equilibrium.
But why inflation creates more time?
The reason is that fast change in the scale factor. Since at the beginning the scale factor was too small but then increases fast so when we take the integral to calculate the conformal time we get a large value.
$$\eta=\int d\tau /a(t)$$
or
$$d\eta/d\tau=1/a(t)$$
Source:http://www.damtp.cam.ac.uk/user/db275/Cosmology/Lectures.pdf
As I said before. The inflation creates more conformal time. You may refer is as "slow". But the proper time is in short period of time.
From the book of Andrew Riddle, Cosmological Inflation and Large Scale Structure
The point is theres difference between conformal time and proper time. Maybe this creates confusion.
Also you can read the article in the link (Chapter 2)