I've found them to be common in the clubs I've belonged to over the years. The Saturday Morning Breakfast Ride is a long and venerable tradition. Sure, your muscles get cold and a little stiff while you're eating, but that clears up the first few minutes after resuming the ride. As far as breakfast sitting on your stomach as you begin the return trip, well, we're all grownups, and we're responsible for what we eat. If you know your body can't handle riding with a three-egg omelet, hash browns and a short stack in your gut, don't eat it. Get some oatmeal, fruit and juice instead. Or don't go.
And as far as these rides being more social or non-performance oriented, that's not been my experience. With the club I belonged to in central Illinois, the only riders who showed up for these were relatively hard-core. And here, the breakfast ride on my side of town is routinely done by two well-attended groups - a slower one on a short route and a faster one on a long route, meeting up for the meal at the same time. Then returning accordingly.