I can understand your reservations, „running a coffee making cycle“ makes it sound as if it is some kind of machine - which it isn’t. It’s one of these percolators, made of stainless steel, that you put on your stove to make coffee. Look up „Bialetti Moka“ and you get the idea, it’s really quite simple.
The reason why the dishwasher takes so long is that it takes time to break down the grime. Same in the coffee maker. There is a deep crevice in the coffee compartment that cannot be cleaned mechanically because you can’t really reach it (clearly a design flaw if you ask me). But remaining coffee tends to build up there and over time it affects the taste. Using detergent and letting it sit overnight breaks down this oily residue and leaves it shiny as new. Then you take the whole thing apart, gaskets and all and thoroughly rinse it. If I put the coffee maker into the dishwasher it would be the exact same chemicals (ok, at 70 instead of 100 degrees Celsius) plus less thorough rinsing. But no one would object.