[T. O'Heir]

"...get a semi-auto or..." Nope. Dry fire practice cycling the action. Got so I could fire 7(mag extension) aimed shots, at plates and pins, out of my 870, in under 5 seconds. That's slow too. The winners were doing it in under 3 seconds. Cheaters were using target loads. snicker.
Mind you, Sporting Clays isn't a speed shoot. Dry fire practice does make you far more familiar with your shotgun just the same.
"...Should I have considerable tension pulling..." Nope. Firm grip, but not tight and cycle the action under recoil. Takes dry fire practice to get quick. Dry firing will not hurt your shotgun.
"...it will not cycle until you release that pressure, even after it has been fired..." Nonsense. How hard you pull a 500 into your shoulder with the trigger hand(definitely not the fore hand) has nothing to do with cycling the action.
"...Winchester 1200/1300..." Slightly longer stroke than an 870 too. Had one or the other loaned to me after my 870's barrel split(plain barrel cut to 18 3/8") at a plate shoot. Fortunately, I noticed the split before I shot again. Bang, stroke, nothing. Short stroked it. DOH!