Sounds like your caliper is just too loose at the moment, hence moving from side to side when the dual pivot is actioned...
The bolt through the fork is tight, there is no wear at all as the bike was unused,
from what I see the complete calliper is supposed to freely rotate on the centre
pin, this action is triggered when the brake lever is pulled, there is a spring
on the calliper that is supposed to push the callipers apart when the brake lever is let go,
one pas flies out when the brake is released, the other one just goes out a little,
when the brake is released, but if I reach for the calliper I can pull the other pad out
away from the rim, or centralise the calliper with my hand, it only takes a little
force to rotate it to the centre position, it seems to be binding a small amount
on the centre bolt that holds the calliper to the fork, the back calliper is the same.
I have no spray oil so will get to it tomorrow.
Am I missing something, as you say it is too loose, but I can’t find any play.