What you are describing is a different sound resulting from ultra tightened thick kevlar heads. Those drums had a single set of snares under the bottom head. The double snare concept is primarily used in Scottish and Irish pipe band drumming, resulting in a very crisp and articulate sound.

Crisp and articulate sound is much more desirable (to me) than than the sound of a pack of dogs with untrimmed toenails running on a wooden floor.

When he was in his high school marching band 15 years ago my son's Yamaha marching snare had a kevlar head. The drum key was a crank so you could really tighten the heads up. With the snares off it sounded like you were playing on a linoleum kitchen table. Unfortunately, even for pipe bands today, the sound is tending in that same direction with the latest snare drums. But, a double snare Premier pipe drum from around 30 years ago was a joy to listen to.

I have a Premier HTS 200 model pipe band drum that I used in pipe bands about twenty to thirty years ago when that version was the current model. It has a white coated Premier Kevlar head but we always tuned the drums for maximum snare response. I have played it a couple of times at drum clinics where there were modern marching drums of almost the exact same mechanisms (Yamaha) but had synthetic gut snares. It was fun to hear the difference. The pipe band drum still sounded like a snare drum (but with high pitch) and the other marching drums sounded to me just like the images you have described so well above. I have not heard in person or played modern pipe drums so I don't know what is their current trend for sound. If you want to hear some amazing drumming, check out anything with Jim Kilpatrick on YouTube.