We have to use screens on a passive solar here in WA state, where we are not particularly known for having lots of sunshine. In order to extend the useful season of the passive features, you have to defeat peak Summer sun, particularly the late afternoon Westerly radiation when it starts sneaking in under the overhangs. Screens are the best way to do that.
During the cooling season does the screening on other windows not in direct sun provide any heat rejection benefit
No. It is only the direct radiation that they screen. There may be some situations in which a window is not receiving direct solar radiation, but gets a lot of scattered radiation, but by and large, no.
simply to maintain 'look and feel'
Why buy additional screens? I'd only use them where they are needed.
Do the screens provide benefit in the heating season too
Not really. There would be some air film modification, but the loss of Wintertime solar radiation would be the worst thing. See cschmelz's comments above. You really do want the roll up/down variety. Even if putting them up or down is a seasonal thing as opposed to a daily thing. Some of ours are even motorized. The house decides if it can get more radiation or needs shading.