Is it safe to remove the rings on jars for long-term storage of home-canned goods

If you are seeing this effect after the jars have been in storage for a long period, do not eat the contents! This is a sign of botulism due to improper canning; the bacteria often (but not always) produce gas as they grow spores.

If this is happening immediately after the canning process, it is probably because you are not creating a proper vacuum seal. There are three accepted methods for doing this: Thermal exhaust, mechanical sealing (i.e. using a chamber vac) and steam displacement AKA steam injection. See the link for more information; the second two require specialized equipment, so thermal exhaust is what's normally done in a home setting.

In the thermal exhaust method, you get the contents very hot (71-82° C), which causes them to expand and release gases (air and carbon dioxide). After sealing and cooling, the subsequent contraction will create a vacuum seal. This is the most common method of home canning.

If you dump in hot brine and immediately seal the container, you are doing the opposite of this. Since the heat takes a while to distribute, you are causing this initial expansion when the jar is already closed, and this will force more gases up into the headspace and probably pop or warp the lid; if you're unlucky, it might even break the container.

Is this method safe? No. How unsafe it is depends on the acidity. Low acid foods must be processed in a pressure canner. That includes the majority of vegetables and especially garlic, peppers, etc. The main reason is botulism; the anaerobic environment is perfect for the C.botulinum bacteria, but spores cannot grow in high-acid environments with pH < 4.6. If the food is high-acid or has been acidified (i.e. pickled) then you don't need to worry, but with low-acid food you must kill all the bacteria and spores, and botulism spores are extremely heat-resistant.

In fact, they are so heat resistant that you cannot even kill them reliably in boiling water. That is why you need a pressure canner, to get the temperature all the way up to 121° C / 250° F, and you need to hold it there for at least 3 minutes. That is the only way to safely can low-acid food. Simply boiling it isn't good enough, and pouring in some boiling brine definitely isn't enough.

For high-acid foods, such as jams, a hot water bath is OK. This still involves boiling the entire jar after it's been closed, but you don't need a pressure canner. Pouring boiling liquid into a jar of cold or warm food will still not get the food up to a sufficient temperature, and even if it did, you would still need to sterilize the jar itself; that's done by boiling the whole apparatus. In this case you're not worried about botulism (since it can't grow in those conditions), but you still have to kill the other kinds of household bacteria, which don't have heat-resistant spores.

Since canning is intended to preserve the food, i.e. for long-term storage, you have to be a whole lot more careful about bacteria. You can't leave any opportunity for it to grow. Lots of people can the way your wife does and don't get sick, but it is a risky proposition, and I would strongly recommended you use safer methods, especially if guests or children are involved.

That passage is suggesting an entire canning/processing method, one that may not be safe. It will generally work to create a seal, as they say, but it may not fully sterilize the contents and the seal will not be as likely to hold. I would not follow their instructions, and instead process your chutney according to a trusted canning recipe you find elsewhere.

See for example this USDA source, which says:

Some other methods of sealing jars call for inverting a closed, filled jar of hot product for anywhere from thirty seconds to one hour. (Inverting is turning the filled jar upside down on its lid.) While this inversion process can be successful in producing a sealed jar, it works best with very hot product. Individual variation in practicing this procedure or unexpected interruptions can result in delays between filling jars, getting lids screwed on, and inverting the jars. If the product cools down too much, the temperature of the product can become low enough to no longer be effective in sealing jars or preventing spoilage.

When the inversion process does work, the vacuum seals of filled jars still tend to be weaker than those produced by a short boiling water canning process. A larger amount of retained oxygen in the headspace may allow some mold growth if airborne molds contaminated the surface of the product as the jar was filled and closed. More complete removal of oxygen from the headspace also offers some longer protection from undesirable color and flavor changes with some types of fruit products. A weak seal may be more likely to fail during storage.