There's a bouncy ball in a box with pressure sensing walls. Each wall feeds the pressure data to HDD physical storage. The ball is released from the top wall at some random angle toward the bottom. The storage devices spin at 100prm and in sync.
Looking at the data from the 6 storage devices, I would see the first entry is from wall_ceiling, recorded at plate rotational angle 0pi. Then the next entry in the collection is from wall_floor say at 1pi. Then wall_left at 1.5pi then wall_ceiling again at 2.1pi, then wall_right at 3pi.
The ball-sensor-storage system alone never needed a "flow of time". The data printout at the end is just a piece of paper with a few columns of numbers. A function is also a column of numbers. And so is a circle or any other polygon.
You may think the printout went from 1 entry to 2 entries to 3 entries...that surely was a flow of time. No. A flow of time is the subjective feeling of a continuous you "flowing" through time. It is entirely internal. The printout system does not have a flow of time. It has time, but again, Barbour is arguing time is extra baggage which is unneeded when you have the motion of the particles - as above. It adds nothing. At each step the printout is entirely given by the behavior of the other particles in the system. Why not our brain states too? When you see "blue" you are seeing 480nm wavelength photons hitting your retina and cones. When you feel continuous, each brain state of yours is kind of a running average of your previous brain states (memories). Or something like that. Maybe some kind of recursive build up of memories at each instant.
But you say you must "flow" even for brief moments to perceive something. Even if it's in discrete steps that's fine with you. But you have to be willing to go one step further to get their arguments. There is only assemblages of particles. And somehow certain assemblages lead to self-reference and awareness, of which we call conscious experience and "flow of time".
Can a function "feel" something? Who knows. But the course of human scientific endeavor has taken us very far from where we started.