I am not an expert on photograpic chemistry but I am a chemist. I also have been a photographer for 60 years. I have thus tried many things. PE could probably provide some more insights on what is happening. But here goes.
The principle advantage of a staining developer is that it produces not only a silver image but also a stain image. Compared to a conventional developer less silver needs to be produced to provide the same density and contrast. The stain image is grainless. Becuase of this the overall effect is a less grainy image. Less silver equals less grain. However, if you like dense negatives then this advantage is lost.
During development a stain is produced in direct proportion to the amount of silver that is reduced. This creates a stain image superimposed on the silver image having similar characteristics to the silver one. The stain is produced from the oxidation products of certain developing agents like pyrogallol, catechol, and hydroquinone. In order for the stain to form the sulfite content of the developer must be low just as it is for color developers. The stain consists of what are known as condensed polyphenols or humic acids. These compounds are highly colored and the stain is permanent. Once formed the stain is no longer effected by either the acid or the sulfite contained in such solutions as fixing baths. Humic acids are only soluble in concentrated solutions of either sodium or potassium hydroxide.
Now both tannin acid and the stain are condensed polyphenols. Tannic acid has a lower molecular weight than the stain and so is soluble in water. I think we are all familiar with that fact that animal hides can be tanned using tannic acid to make them stronger. A staining developer does the same thing. The chemical collagen in both the gelatine of the emulsion and the animal hides can be tanned thereby hardening them. Some people say that this tanning action prevents the silver grains from migrating and clumping up thus reducing the grain of the negative. With today's prehardened emulsions this alleged benefit may not be as great as in the past when emulsions were rather soft.
The tanning effect can be seen by the naked eye as it causes the emulsion to shrink producing a relief image. This also produces a refractive effect upon enlarging.
If the silver is removed from a negative produced by a staining developer the result is a grainless image similar to that produced by color films. Like so many things in photography this can be good or bad. Yes, there is no grain but the human eye may not perceive this image as being sharp. Anyone who has seen large color prints will experience this effect.
The amount of stain produced varies with the choice of developing agent. Some produce more stain than others. The color of the stain they produce may also be different. The developing agents differ in the conditions under which they produce a stain. Pyrogallol acts like a regular developing agent very similar to Metol in the presence of moderate to high amounts of sulfite producing no stain image.
I hope this brief description answers some of your questions.