I seem to be failing at communicating the important points. So I'll try again :
Torque in of itself as a number (or area under a torque curve - by definition a single number, the numerical integral of the torque curve) as a figure of merit of an engine is utterly useless without information as to the rpm at which that torque was measured. Two motors with identical "area under the torque curve" might have wildly different performance in a given chassis. You must specify the rpm at which the torque is measured to have any idea how "strong" a motor is. Conveniently we have this thing called power that does that :) Still can't see the reluctance to use it!
If you specify a motor makes 500ft-lbs - in theory it could be any possible horsepower. If it makes 500hp you know how fast it will accelerate your chassis at any speed (-drag of course). You have no clue with just torque. If you include the RPM at which the torque was measured... you effectively are using the hp in the calculation, without ever calling it by its proper name.
I think everyone agrees single measurements of either hp or torque are inadequate to describe an engine (although with a single hp # you can fully describe the performance of a chassis connected to a CVT, with torque you have nothing - hence the better descriptive power of hp), and want to use some number calculated from the curve. The integral seems like an obvious choice. Any time you are going to be taking the integral of either a torque or hp curve, you need to specify the operating range of RPM. In "race" mode the widest possible range is usually the rpm drop from 1st gear revlimit to 2nd gear. When you shift at revlimit in 1st the rpm in 2nd is = revlimit*(gear#2 ratio/ gear#1 ratio). This defines the "range" over which the hp/torq of the engine is relevant to performance. Perhaps the lower rpm should be extended lower than this as its unlikely you will downshift to get 100rpm worth of 1st gear!
For street driving you typically don't revlimit shift, and the rpm range is both lower and in a wider ratio. In this case the integral needs to be performed over this wider range. Motors with narrow peaky powerbands will suffer under this metric as the example several posts ago shows.
Whether the Torque number is derived from the observable HP on a dyno
or not, I think it's a more valuable number. Since the HP number is
simply the amount of torque multiplied by the rpm, a number that
distills the area under the HP curve is going to tend to show more
favorable numbers for engines with less torque that rev longer. (say an
S2000)
The S2000 motor will have a excellent hp integral over the ratio of rpm used in racing. No surprise, this is an awesome motor. It will have a decent number even over the wide rpm ratio. The problem isn't with the S2000 engine, its with the S2000 chassis, its too heavy for a "small" motor. That motor in a 1800lb chassis would be considered more than "torquey" (actually people would wet themselves over its torque). However a larger displacement slower-revving motor could easily have a higher hp integral.
Performing the integral over a ratio of rpm is superior to performing it over actual rpm due to the speed effects described a few posts above. For example performing integral from 2000-4000rpm is wider (in speed) than from 4000-6000rpm, despite covering the same # rpm. The ratio is the important information as that describes the speeds covered in each gear.
Any single number that describes the engines low-speed performance is not really applicable to racing, and any single number that describes the engine in the range used while racing isn't descriptive on how it is on the street. There is no way to "compress" the information into a single number that does both. The hp integral over ratio of rpm "of interest" however is a decent figure of merit for an engine, either street driving ratio (large) or racing (as narrow as your gearbox).