I hardly ever used workers, if any, I built walls with them, but I did not seem to have a need for their food carrying services more than once in a whole campaign (and that more because of impatience rather than real need of it).
I think you summed it up. It was really only used by hard-core micromanagers when they had trouble getting the supply line system to do what they wanted. Workers and slaves in Rome are used strictly for resource production similar to how you would use them in other RTS games. Your primary decisions will be whether to use your limited recruits as workers to gain more resources or to train them as soldiers for offense/defense.
But this sounds good. If a heavy march unit is attacked, can it just make a big pile with its excess food and rearrange into fighting stance (like those piles slowly fading away after a surrendered unit left them behind) or do you have to build a supply camp / fort to dump the extra food into it if you don't want it to be lost?
Your unit won't drop their extra food unless you specifically change their stance from "heavy march" to something more combat oriented like "battleline" or "skirmish". While they are carrying extra food they can still fight but they suffer a significant morale debuff. If you do choose to drop their food then the food is permanently lost but the morale will not immediately return to normal so it will likely be too late if you are already in combat.
We decided on this approach because we didn't want to encourage players to babysit their units and quickly switch stances at the last minute. This way you'll want to decide whether a heavy march is the best tactic in advance based on the likelihood of an ambush and, if you think the risks are high, then you might want to group your units with others who are still in a combat stance. Your unit AI will also use the stances to determine how to react, so if you've got multiple units in your group then the ones in combat mode will try to protect those carrying supplies.
Are there other strategies for the barbarian factions or do they also build supply camps?
All factions can build supply camps. While the Romans were obviously the most famous, German and Gallic generals like Ariovistus and Galba both built camps to support their military campaigns.