Question: why is this a problem? What is wrong with extra sensitivity?
Because the instrument not only overreads under g, it suffers a lot of lag and it's hard to find the wings level position under g loading.
Question: can this really be entirely correct? Why would the direction of rotation be reversed?
Beacuse the turn coordinater is designed to be both yaw and roll sensitive with emphasis in greater sensitivity in yaw. This is to help the pilot coordinate his turns - something I couldn't really see in the very few times I've looked at one. I think it was more of a marketing thing.
Question: in practical terms is it in fact the case that the turn coordinator is much less useful than the turn rate indicator, for the specific purpose of partial-panel recovery from unusual attitudes? What are the characteristics that make it so?
Yes, because of the reversed rotation, the turn coordinater under reads under g. This makes it hard to find the wings level position.
AFIAK, the T and S rotates away from the pilot, the turn coordinator towards. There may be manufacurer's variations, but this is the case with the instruments I am (was) familiar with.
To understand how it can effect a recovery, I will describe the partial panel UP recovery we used to teach in the RAF.
1. Check altitude - if below MSA - eject.
2. Unload the g. (To do this, push to 0g, then check back on the control column a little).
3. Roll the aircraft to centre the turn needle. Anticipate about half a rate for lag.
4. Pitch the aircraft until the altimeter needle stops, then check the other way (to account for altimeter lag) a little.
5. Repeat roll and yaw inputs until the aircraft is straight and level. (These should be progressively smaller and you should only need about two or three,)
6. Re-erect the AH.