I worked in the offshore oil industry for a few decades. We usually painted galvanized grating and handrails in the splash zone to extend the life of the steel tremendously. The splash zone is that area near the water line where extremely corrosive salt water splashes against the steel and will corrode it very quickly. I've seen galvanized grating where some of it was painted because it was near other steel when it was painted in place and some of it was not painted. The difference in just a couple of years was easily apparent.
The galvanizing we used was commercial grade hot dipped galvanizing, not the galvanizing that is sprayed on. The paint was industrial grade with good primers and intermediate coats. We used a weak acid solution to clean off the surface and get rid of that powdery surface layer that can develop on galvanizing, then rinsed it well with fresh water to get the acid off. The paint was applied by professionals under certain humidity and temperature conditions. We normally used a 4 coat system, a primer, two intermediate coats and a top coat of urethane.
I am not aware of what quality the galvanizing is on the wire you are using, hot dipped or sprayed. Surface preparation is important to get the paint to stick and last. The main thing is that it is clean, not greasy or with anything loose like scale or dirt.
A big problem with coatings, whether galvanized or painted, is holidays. Those are holes in the protection, whether pinholes through the galvanizing or painting, or scratches. Corrosion is going to be concentrated at these holidays. I'm not always real good at getting fencing up without scratching it someplace.
I seriously doubt you are going to get a professional quality paint job on your fencing, either in materials or surface preparation. I do think if you paint it, taking a bit of care that it is clean, that you will fill those holidays in the galvanizing and provide a mechanical cover for the fencing to take some of the abrasion and help minimize scratching the galvanizing while you install it. I think you will extend the life of your fence quite a bit, especially the part in contact with the ground. I'm not a painting expert but I did work closely with the guy in charge of painting. Often I'd repair the grating or handrails and he would come immediately behind and paint it. His was a specialized field.
With all this said, I did not paint my run fencing for corrosion protection. I did not see it as cost effective in this application. I did run the roller over a lot of it after I finished painting my coop dark green so I could see through the fencing better from a distance. In the extremely corrosive salt water atmosphere offshore, painting the galvanizing to keep from having to replace it in a few years was very cost effective.
Anyway, that is my opinion and my experience. Take from it what you will. Good luck!