RDR definitely gained a lot from GTA IV and GTA in the future has a lot to gain from RDR.
I definitely like the bag snatcher idea, they could perhaps even give you the ability to steal bags yourself. You do play as a criminal after all.
Perhaps there could also be police incidents, maybe a shoot out at an armed robbery or something, you can choose who to help. Perhaps occasinally you would see a police car or two rushing somewhere with their sirens on, if you follow them you arrive at a crime scene.
This could also be a way to incorporate some Ambulance or Fire fighter gameplay back in. Say you see a fire truck on it's way to an emergency, you follow it and arrive at a burning building. The fire fighters can try to put out the fire while you heroically/foolishly rush in to save someone who's trapped. This could be a reason to bring back the fire extinguisher. Or you could man the water cannon while the real fire fighters go in to the building to rescue anyone.
Perhaps for an ambulance encounter, you see one heading to an emergency and follow it. When you get there, there are too many people to fit in one ambulance so a paramedic has to stay with the remaining victims. Meanwhile you have to drive the ambulance back to the hospital, then you return the ambulance to the scene of the accident and the paramedics take it from there.
Not necessarily realistic ideas but they might work in a game like this. They don't need to be too complex, just smallish things like the ones in RDR.