You write that you invested the time needed to "locate and identify" the animals.
Did that include visual location and identification?
The goal was to eyeball the animal, and I have frequently succeeded.
Where was the cougar or bear? Not in a tree or cornered?
I have never cornered or treed a cougar. I have, however, succeeded in tracking and watching cougars on several occasions. Usually I am able to find them at choke points, creek crossings, and dens. I have spent as long as a week tracking a single big cat, and not always successfully.
Bears I have found much easier to track, if only because I am not so concerned about them back-trailing me and they are not as light and agile as the cats. In winter I almost always find them at a den-site, other times I have found them foraging. The larger bears are not all that quiet either.
Did your trained or untrained dogs refrain from barking? Or, if not, didn't the barking alarm the subject?
The Otterhound never barked or howled, although she did mutter a lot.

Absolutely the best scenthound I've ever known. She could track scent that was more than 48 hours old. Just her presence was enough to flush a cat that I would otherwise have missed.
The dogs that made the greatest racket were the hunting dogs that we drafted for tracking purposes. I am certain they alarmed the bears.
The trained trackers - a pair of Bloodhounds - belonged to a friend of mine that used them to track lost pets and the occaisional bear. They were great but would get a little bit enthusiaatic when we would get close. They were also of little use if we were going to be out for more than a day or two (they had no "off" switch).
The quietest dog I ever used was a 100lb Bouvier that was no good whatsover as a scent-hound but was very smart when we would get to the vicinity of the animal. He would stop in his tracks and stare right in the direction of the bear. When that would happen I'd know that we were very close indeed.
We did not use the dogs to "run down" the quarry. We used them to keep us on the trail - when there were long gaps between spoor or poor ground for prints - and to warn us if we were coming up on something unexpected. In the case of the cats I always assumed that they were watching us and so I was, well, a mite bit concerned that they might circle around on our back-trail. That was never an issue with bears.