they do. i flown last week from MUC to FRA. A320 LH. And the Pilot started Engine No.1 already at the Gate BEFORE pushback, while pushback he started No. 2.
But it wasn't during baggage loading, and re-fueling...etc. It was most likely at the very last stage before pushback.
GSX allows engine start while pushing back, which is the most realistic option.
In any case, it doesn't make any sense to be able to do this at all.