When *brewing* Irish coffee in a drip coffeemaker or French press, do I need to use more Irish whiskey?
I don't think you'd need to increase the proportion.  You could, but I'd probably advise against it mostly because of sobriety concerns.  It's much easier to over-serve someone with a more potent cocktail than it is a milder one - especially after the first couple.

This is kind of an interesting method, and I think the key to success is the gentle heating of the whiskey over the heating element before the coffee is dripped in.  Ethanol is pretty potent stuff (ever seen a Molotov Cocktail in action?) and will evaporate more readily than water will.  This is why when adding wine, vermouth, etc. to a dish, you can drive off most of the alcohol at stovetop temperature well before the water evaporates.  Adding the whiskey to a warm carafe is likely driving off a small percentage of the alcohol, reducing its bite (and its punch as well, but this seems like a reasonable tradeoff for something generally considered a dessert drink).

Some of the depth of flavor could also come from volatile flavor compounds (things like esters which develop as spirits age) being driven off or changing composition.  Heating spirits or setting them on fire isn't a common trick these days, but historically bartenders often used it to transform how the flavor of spirits is sensed by the drinker's palate.  It's the flip side of chilling and diluting drinks through stirring or shaking, and it's a good substitute for those in the middle of winter, when more ice is the last thing you want.

I think it's also worth pointing out that the brand and variety of both coffee and whiskey will do a lot to change the end result.  Coffee obviously has a huge number of origins and roast levels, plus myriad other factors that can affect its flavor.  Irish whiskey varies too; it's often seen as a monolithic category, but it'll differ as much as scotch or bourbon depending on the quality of the mash, how it was distilled, and the aging process.  Powers is unapologetically brash and spicy; Knappogue Castle is a much more subtle product that would be really well-suited to this type of application.