As you say, mashing in without stirring would not be good.
If you stir during the mash, you may speed the mash up a bit, because you are mixing up the enzymes and giving them a better chance to get to the starches. However, stirring a mash without adding heat is likely to cool it down a bit. Some stir the mash, some don't. It is a matter of personal preference.
At the end of the mash, I usually do a mash out with near boiling water, followed by a fly sparge. If you do this, stirring is essential as you need to equalize the temperature throughout the mash, but you should not stir any more once you start the sparge.
If you are batch sparging, you will need to stir after each addition of sparge water to dissolve the sugars prior to vorlaufing and draining. If you don't stir here, you will get a situation similar to adding sugar to a cup of coffee without stirring. Most of the sugar will not dissolve, and you will get very little sweetening for most of the cup, and most of the sugar will be concentrated right in the bottom of the cup. In the case of a sparge without stirring, much of the sugar would be left in the grains.
As for stirring causing excess tannin extraction, it won't unless you let the pH rise to over about 5.6 and/or you get the grain temperature too hot.
Stirring is OK IMHO, but squeezing the grains to extract every last bit of sugar will cause excessive tannin extraction because of the mechanical action of forcing the water out of the tannin rich husks.
Hope this make sense.