Bone broth provides a host of nutrients which help rebuild the mucosal layer. Things like glucosamine, condroitin, glycine, minerals such as calcium, magnesium, etc.
Yes, it's true, if you have a leaky gut anything you eat during that time can lead to antibodies to undigested proteins in that food if it makes it past the blood-gut barrier. These allergies might go away within 6-12 months, or they may become permanent - in the worst cases their amino sequence resembles some part of our own tissue and it will become an autoimmune disease.
Its possible that the broth itself has aminos that are non reactive depending on how they were extracted - certainly glycine itself shouldn't cause automimmune reactions.
The most important thing is to close those gaps as quickly as possible. One thing most people don't know is that our blood-brain barrier is made of the same cell types as the blood-gut barrier; if you have leaky gut, you'll likely have leaky brain as well.
The problem is a big one, and because grains signal zonulin to open the gaps, they can cause quite a lot of damage. There are other things that can damage the lining, but in the SAD, the most common, most likely source is going to be grains - it certainly can be an infection, or a stressor.
The human body is alive and will heal rapidly, all else being equal. All else is rarely equal, however.
The q of why it takes so long for the gut, which has an excellent blood supply, to heal is a critical one, along with why some people's guts never really heal at all. My personal suspicion is that biofilms of the wrong kind of flora establish themselves and are damn near impossible to boot outta their entrenched positions.
Perhaps bone broth helps by neutralizing the little critters somehow? Or perhaps even if it leaks, its restricted set of proteins does not contain any that are seen as foreign and deserving of a major immune response?
Yes it could potentially leak through the intestinal lining. The idea is that bone broth and supplements like L-Glutamine will over time form enough of a barrier that less and less stuff leaks through. That is why leaky gut can take months to heal -- you're taking 2 steps forward and 1 step back constantly trying to heal it.