Sir, are you sure that your infantry were on Travel Mode ('T' key)? You have to be on Travel Mode (a.k.a T mode) in order to utilise roads.
It seems that you know that your machine gunners have to be on T mode to move, but turning them into T mode uses 1/3 of their Movement allowance on that turn. Moving them on the same turn as they changed the mode isn't a good way to see their speeds. I am not sure whether those were on T mode or you changed them to T mode on that turn, however.
I don't have East Prussia '14, but I have France '14 and I checked infantry speed on the road. They can go 6 hexes per turn in T mode on primary roads.