I agree with you 100%. It really does "feel" like something good is happening when you do some running first. And I do sweat a lot when I start workouts this way. In high school I played sports, and in every practice, regardless of the sport, when ran or did something to get going before we started more specific sports stuff.
I kinda feel more satisfied with my workout if I'm sweating a lot. Ridding the body of impurities is good for you.
It's difficult to lift after running for the first week or two, but then you(or at least I) get used to it REAL fast. There has to be some kind of benefit there. Also, once you get accustomed to it, its like the best warm up.
Running/Cardio is not as much fun to me as lifting. If I run first, I can't skip it.
Back in the Dark Ages (pre-2000) they used to tell us that you'd start burning fat after 20 minutes of exercise. Regardless of the specifics, 20 minutes then fat burn. I feel like this run first set up was pretty common back then.