Ahoy,
I use my running backs anytime I am in the ocean,as I am on my way somewhere. Whether up wind or down. Here in Portland, which is on a river, I do not use them and, in fact, they are physically removed from the boat.
When going to weather, if you sight up the mast, without the running backs, you can easily see the mast "pumping". The running back will easily stop this movement. In doing so, it slightly pulls the inner stay back and tighter. At the same time, this adjustment forces the mast head upward (for sure very slightly) which also tightens the headstay.
Going downwind, the running back is one more stay to hold things up and just seems like a good idea.
Concerning the barber-hauling of the staysail: My staysail is on a boom and controlled by a single sheet with a track and movable car. I normally have the car "stopped" about one foot off center. When reaching, I can run the car out as far as the cabin edge. If using the staysail on a broader reach or run then I will vang the boom down to the forward hawse-pipe. Nothing fancy here.
Good Luck, Dave