why use a pressure cooker for collagen to gelatin conversion?

What you need for the conversion of collagen is a certain amount of energy. It is a complicated process - the melting point is around 70°C for the type of collagen contained in beef, but the melting does not happen instantly once the meat reaches 70°C. In a pressure cooking, you can apply the same amount of energy in a shorter amount of time. This is not bad, as opposed to slow roasting of collagen-poor meat.

In collagen-poor meat, you have two types of protein, which are soft and wet. Under heat, they curdle, becoming tough and dry. The perfect meat is when the first type has curdled (so the meat is not raw) but the second hasn't, so it still holds juices inside. If you curdle both, your meat gets tough and you can't take it apart with your teeth.

In collagen-rich meat, you curdle both proteins - the collagen itself is tough and you want to melt it, but this happens long after the meat has curdled. But because the muscle fibers are not clinging to each other, but separated by collagen, you still get tasty meat. For that, you melt the collagen into gelatin, and serve the meat warm, so that the dry fibers are separated by the smooth, juicy melted gelatin. Unlike slow-roasted meat, you don't have to tear the juiceless fibers apart, and the gelatin makes up for the missing meat juices which were expelled from the cells during curdling.

So, in slow-roasted meat you don't want to cross the temperature limit for curdling a certain protein, this is why you have to apply heat slowly until the center of the meat has cooked, without the outside getting overcooked. In collagen-rich meat, there is no upper limit at which the meat gets non-tasty, so you can push the energy needed for the collagen-to-gelatin conversion quickly into your meat. The pressure cooker can do this better than the normal boiling process.

This is something I feel confuses a lot of people when talking about sous-vide cooking. I think the best way to think about sous-vide is as "low temperature cooking". In these temperature ranges collagen breakdown can still occur but it takes significantly more time than it would at higher temps in a traditional braise or in the pressure cooker where, depending on your elevation, water can boil at 121C.

The real answer depends on what your goal is. If you want the texture of a traditional braise you could cook something at say 85C for 8-10h and it would be fall apart tender just like in a pot in the oven or the pressure cooker. Now if that was your goal I would say save the time and do it another way. However, if your goal is to augment the texture of a food through SV cooking then cooking for a significantly longer period of time at a lower temperature will give you textures you cannot achieve any other way. So a short rib cooked at the 85C/8-10h will be falling apart like mom's old school pot roast but if you went lower at say 54C/72h it would be as tender as a tenderloin cut but with an intense meaty flavor and beautiful medium rare color.

So for a recipe like Alton Brown's chicken and dumplings he wants to accomplish this conversion as quickly and efficiently as possible and is likely using the collagen rich chicken stock created from the process in the soup so that's why I would turn to the pressure cooker. Plus, he's a TV chef so he's not going to be recommending non-mainstream cooking procedures like sous-vide at that point in time.