I know, it's terrible. I share the same frustration. I usually leave with only hand luggage, have egate, and use the J/F Class separate check-in area. From home (Marina) to lounge 60 minutes max, usually more like 45. So I leave around 90 minutes before departure, which usually leaves enough time for unforeseen delays (traffic on SZR), breakfast/lunch/dinner at the lounge, at a comfortable boarding.
Since "they" are unwilling/unable/indifferent, the only solution is to make the driver wait. He calls you when he arrives, just tell him that you'll be a little longer. You may get another call from the EK dispatcher with all kinds of silly warnings and telling you that it's at your own risk. Just say yes yes and leave at a time that you think makes sense. I feel a little bad to make the driver wait, but then several of them told me that they don't give a toss about whether they wait at the airport to drive someone to a surprise destination or whether they wait at a pickup point or whether they drive. All the same to them.