Is there any issue with (water-based) liquid emerging from a piece of meat which is being grilled, this liquid dripping onto the burning coals? I have a gas barbecue and a steak regularly drips onto the naked flame.

The dripping of the meat on the coals is part of the cooking process. It is not that you are intentionally extinguishing the flame. The Shulchan Aruch (OC 507:4) writes that when that when the extinguishing of the coals (or in your case the flame) is part of the cooking process it is permitted the same way the cooking itself is. The Shulchan Aruch (ibid) further specifies that placing meat directly on the coals is permitted even though when first placed it would definitely extinguish the coals a bit.