I've noticed recently that the contestant who is picked is already miked up when he/she is shown reacting to their being named as that day's player.

This is a change to the previous practice when no players were miked up until after the reveal.

Surely more than one player would be wearing a mike at the top of the show or the name of that day's player would be blindingly obvious to the contestants if to nobody else?

They changed the process a while ago. The player is told they are playing whilst they are still in their holding area before they come into the studio. Hence the slightly less surprised expressions on their faces now.

Been like that since October 2011

...but only on-screen since the start of 2012.