squeak- the water must be heated- never boiled- BEFORE they drip over the ground coffee.There is no way of making fresh coffee- in a machine- with cold water. Possibly in the old times , they just threw ground coffee into a pot of water and then boiled it-ech!!! there is very littel worse than bolied- burned coffee! unless it is instant coffee….
BTW- ground coffee is more like “ofui” (baked-roasted) and hence it is “bishul achar afyiah”.
I briefly looked over some of the halochos concerning “shehiah’ , “chazoroh” and “bishul”. BTW- the fear is not to stir the pot (that would be “meigis”, toldoh of bishul) but the fear is that you would shake the coals (or other combustible) under the pot (“mechate es hagecholim)to allow the food to cokk better or quicker.
I am not sure whether any of the halochos mentioned can be applied to a coffee maker. Simply, because, there is no fire at present, at the minimum it must be considered like “gerufo uketumoh” , an oven whose coals have been taken out with all its subsequent “kulos”. I don’t know how it could be “shei’eh” (allowing the food to stay on the fire) or ‘chazoroh’ (returning ther food over the fire) maybe it comes under the heading of “chazoroh’ by a non -jew and see whether this applies to an inamite object. Thanks to all the contributors.