I don't know how to answer this thread without stepping on some toes. But I'm going to try anyway. First off I love my lever guns. From my little .357 up to my big old .45-70 (of which I have 4 of). But I also like those flat shooting cartridges and those high end bolt rifles. I just got a nice Bergara in 6.5 CM that I'm tweaking loads for making it a 1/2 MOA shooter. I'm there now but still have some more testing and having a blast doing it. Hunting here in PA I'm usually packing one of my .45-70s or a muzzleloader. My shots rarely go beyond 100 yards and usually from a sitting position or offhand.
Hardcast I bet that those guys you "witnessed" shooting at game with their very accurate rifles either had one or two things going against them. They probably either never practiced shooting their rifles from field positions or they had a case of buck fever - or both.
I recall an old saying "Beware of the man with one rifle, he probably knows how to use it." I really don't know how many rifles I have. But I do know that I know how to shoot them all. Because I practice with them a lot. Not as much as I used to. But often enough to keep sharp.
As for shooting running game, I used to do that in my wild younger days. But at my age I don't need to kill anything to have a successful hunt. The only time I will shoot at running big game now is if I wound it. I can't abide someone who hits an animal in the vitals and watches it stumble around just so they can say it was a one shot kill. Shoot the poor thing again. Give it some respect.