Xjyuk

It has been suggested to me in a comment, about an answer of mine that suggests using a hip belay, that:

There's a reason ATCs, plates, GriGris, etc got popular: they're safer.

I know I stopped using hip belays when I got a proper harness and realized how much more comfortable belaying with a Munter hitch was. I didn't even consider that it might be safer. While I agree that GriGris and other devices with assisted braking capabilities are nominally safer than a hip belay, has there been a climbing accident attributed to the use of a hip belay that would have likely been prevented by using for a belay device without assisted braking capabilities?

It not necessarily unsafe, it's just fallen out of use with better gear and techniques. It still has a couple of use cases and it's something I think people should know, but I would never use it for rock climbing unless I had no other choice.

The use cases where it still makes sense is when belaying another climber from above in terrain that could lead to a slip but is not steep enough for actual rock climbing. In that case, a hip belay is faster to set up and use and there shouldn't be a risk of a whipper. Two clear examples might be going up a snowfield while roped and secondly belaying up Class 3 or Class 4 terrain where full-on rock climbing with dedicated anchors is not warranted.

I have used it before in both those cases and caught a slip but wouldn't suggest doing it for any Class 5 rock climbing (especially from below such as at a climbing gym.

Here is a tidbit from Climbing Magazine:

Hey, 35 isn’t old! It’s definitely not old enough to use hip belays by default, since the first belay devices came out in 1970. The hip belay is a good technique to know for rolling but still “no-fall zone” alpine terrain.

This states that hip belays haven't been an accepted default practice since 1970. (Interestingly as you will see below this date might show how Europe was more advanced than Europe in the field of climbing safety.) Next the article clarifies that hip belays are a good technique for low angled alpine terrain. That doesn't describe modern rock climbing in the context you're advocating for using hip belays.

In the book Accidents in North American Climbing 2016 they have a section about belaying technique and the history of advances in belaying technique. They start by explaining that the earliest belaying is simply a person hanging onto a taut rope for dear life. Then people started to realize that's a terrible plan so they leveraged friction Here's some snippets from that book (emphasis mine, also please pardon the liberal use of ellipses I retyped the text from a scanned image of the book. If you follow the link you should see the full text.):

The addition of friction to the belay system allowed smaller belayers to secure bigger climbers. Wrapping the rope around features in mountain terrain or the belayer's body provided enough friction to hold larger loads.

...

Since the addition of friction to the system, every major evolution in belaying has involved some sort of technology. First came the carabiner, which not only allowed belayers to augment their friction belays but also invited the use of hitches, tied to carabiners, as belay tools. The most effective of these was the Munter hitch.

The Munter hitch offered a breaking position that was the same as the pulling position, so the belay cycle was easy to teach and learn. It soon became the predominant belay technique in all disciplines. (Before the advent of reliable protection, dynamic belays, and nylon ropes, belaying was primarily the duty of the leader. A second might belay the leader, but the leader was not expected to fall, nor was it widely expected that a leader fall could be caught.) ...

THE MODERN ERA

... However, by the Second World War, climbers began to use nylon ropes and other equipment that could handle the forces of leader falls. ... Pushing the limits of difficulty also became more common-leading to more falling.

Belayers around the world also began to experiment with new belay tools that redirected the braking position 180 degrees-the most common early example was the Sticht plate, but the same principle applies to today's tube-style devices.

As you can see people started moving away from body/hip belays almost immediately. They first used technology like carabiners and Munter hitches to increase the friction and decrease the strength required by the belayer. Then after WW2 when they finally had gear that was reasonably capable of handling falls they ended up creating belay devices that use many of the same principles of modern belay devices. The reason why people moved towards the belay device and away from body/hip belays is the whole paradigm changed in climbing. Initially like the book said:

belaying was primarily the duty of the leader. A second might belay the leader, but the leader was not expected to fall, nor was it widely expected that a leader fall could be caught.

This morphed into people pushing the limits with leaders falling and expecting to get caught by their belayers. In order for this to reliably happen the belayers needed belay devices. I don't think you're going to find modern accident reports pointing to hip belaying a lead fall as the cause of the accident because climbers have recognized since WW2 that belay devices are a force multiplier. As such people don't use body/hip belays on modern technical rock climbs so there aren't accidents to record.

In my opinion I think you need to justify why a technique that was completely abandoned in the context you were using (modern technical rock climbing) is safe. The climbing community is primarily focused on two things, safety and praticality. They still embrace the hip/body belay in low angle alpine environments when a leader is belaying a follower in a no fall zone where falls are unlikely. If the technique was as safe as a belay device then you'd see it recommended for modern technical rock climbing. Indeed the first improvement on the body/hip belay, the Munter hitch, is still taught as a viable modern technical rock climbing belay technique. Climbers are endlessly practical as long as the result is safe. Body/hip belays simply aren't safe for modern technical rock climbing.

Original Answer

A hip belay will never be able to have as much security and holding power as a belay from a device or munter hitch. With a hip belay you can never get as much friction on the rope because the minimum radius is the size of your body. A munter hitch or proper belay device can practically fold a rope 180 degrees. Another strike against a hip belay is you need to use more strength to maintain the friction. I can comfortably hold a 200 lb weight using a belay device like an ATC attached to my harness with one hand. I wouldn't be happy doing the same using a hip belay. Furthermore I the unfortunate circumstance where the belayer gets lifted up then a hip belay is deadly because if the belayer rolls the wrong way the hip belay turns into a person holding a rope behind their back.

Finally there's no escaping a hip belay. With any kind of belay device you can rig a reasonable escape. For example from an ATC you can free your break hand by wrapping the rope around your leg. That allows you some freedom of movement and frees both hands to build an anchor. When you're holding a hip belay one hand must be firmly committed to maintaining the belay and your body must remain relatively immobile because any motion will increase the load on your break hand.

Hip belaying a leader while rock climbing is dangerous. Just like pounding a 2x4 into a crack. Either one might hold a fall and both have been used successfully but neither one should be considered safe. Today we have much safer options. Carabineers and belay devices are cheap insurance.

Hip belays can be appropriate for low angle falls especially when belaying a follower. This is because the forces are low and recovery after a fall is quick. Furthermore the chances of a bad fall in these situations is so low that the additional risk caused by a hip belay is more than offset by the increased speed.