I agree with your premise. "Training" isn't just training the body. As Salazar said, it's also about callusing the mind. Training well is about assembling various element of the physical plant into a coherent and well-performing whole, but it's also about psychological preparation. The mind/body connection being what it is, one form that psychological preparation take is exactly what you describe, in my experience: speeding up when you want to slow down.
Getting back into shape, this past Sunday I did a 10-mile long run--which for me is a short long run--and in miles 8 and 9 I pushed things to marathon pace+ which for me is 85-87% of max HR. That's a comfortable sub-threshold pace once I'm warmed up. So I was there. In the final mile, I kicked it up and, very much in line with Kellogg's teachings, I found that I was sailing along at a HR of 95%, which is 5K effort, but was completely in control, not panting, just working hard and well and exactly where you'd want to be at that % of max.
THAT is where you want to be. I really felt as though I could have kicked it up another notch or two. So I sailed home like that.
It wasn't a formal tempo run. It was a progression run that served some of the same function as a tempo run. The key thing was the subjective feeling of being completely in control.
Now, when I do 3 mile tempo runs, they rarely have that feeling. I warm up 5 miles EZ, then do the tempo run. And I almost always find myself picking it up in the final mile, but I almost never feel as I felt in that 10-miler.
I haven't quite addressed your question, but perhaps I have. It's always a good idea to seek the place where you're blasting along, perfectly coordinated, not straining, yet really hitting it hard. That is NOT the same as pushing the pace in a tempo run, where your breathing was already slightly over the edge and now you're just adding insult to injury.
Just "pushing harder," by itself, means nothing, and is probably not a good practice in tempo runs. But significantly upping the pace and effort level at the end of what was otherwise a good, honest, but not over-the-top tempo run or progression run is almost always, I think, a good idea.
Just not every week.