Many thanks Azimuth and Geoff for the guidance/comments which have been helpful. Here I'll do a quick roundup to summarize things in case anyone wonders what seems to work in practice
I found that the best motion is to wedge a small/thin flat point screwdriver into the top link in one steady and smooth motion (should feel a little like when you roll the edge of the blade of a case back opener around the edge of a tight case back - the force exerted by the blade in a forward motion wedges open the link). This works better for some links than others. You can see in the pic below one of my links which warped when it had to be pried out as the two halves of the links were clamped together very tightly. When reinserting into the band, just cover the link with a svelte cloth and use pliers to firmly press back into position
As mentioned above, this procedure can wear down and break specialist jewelers screwdrivers, spring bar tools etc. I found the best (most sturdy/durable) tool to be a Muji pack of screwdrivers which are strong enough to withstand the punishing prying and wedging actions, and also cost a lot less than any specialist tools. I think I paid around USD10 for this a few years ago