The engineer's reasoning has little to do with heat-stressed athletes. That's the quoted engineer, not Jim. People do tend to be able to drink more cool water than cold water when under heat stress. Think about it, can any of you chug lots of truly cold fluids if they don't contain anesthetic alcohol?
Let's round things a little and say that body temp is 100F and you are considering drinking water at 40F or 70F. Water has much of the heat capacity in a body. The other stuff has some, but not nearly as much as water which is one of the unusual qualities of that substance. A person of 167# has 100# of water. If your 167# drinks a pint of 40F or 70F of water, you will lower your body temp by roughly 0.6 or 0.3 degrees respectively assuming total and immediate "mixing" That is much more significant that I might have assumed before looking at it quantitatively.
I am however, disturbed by the lack of detail that I find in discussions by people who project as experts in this respect. We are not just a bag of water and some other stuff being diluted with a little bag of water.
I have a vague recollection of some expert stating that drinking cold water slows absorption. It makes sense that cold water makes your gut cold and slows everything down at best and shuts it down at worst. I just went with that in my previous post, shame on lazy me. Local drastic cooling, however, just slows or stops everything that an organ is doing and that includes absorbing water. In a biology laboratory, we make processes of all kinds stop, or nearly stop by putting them in and ice-water bath. (That said, it is not as effective with poiklothermic cells/tissues/animals.)
I am not trying to refute that "cold stunned gut" reasoning, but I find little data to support it. There is data that supports drinking cold or cool water on the one hand, or warmer (60F) on the other. What it ignores is how the drinking is done. Let's take some likely heat-stressed athletes which would be outdoors meaning ignoring indoor types operating with nice HVAC. You have a very different situation comparing cyclists sipping some COLD water as they want on the one hand and a marathoner or futbaler grabbing a cup or two of ice-cold water and chugging it down in a few seconds when it is offered/available. In the former situation, the cyclists are not going to significantly cool their stomachs. The internal cooling effect is going to be spread out between their mouth, esophagus and further down their guts. In the latter situation, you have that local gut cooling/slowing problem that could be real.
There could be good data out there that illuminates this problem, but I did not find it in a quick search. FWIW, most of what I just typed is just hand-waving as well, but at least some of it is quantitative hand-waving and thanks to Jim for starting us down the numbers path. My science motto is, "If you can't be quantitative F*** it".