It always kills me to hear a beautiful set of drums that probably sounds great to the naked ear, sitting on the drum throne, played live against a couple of Marshall stacks and a nice 4x10 bass rig. The toms end up sounding like cardboard and the backbeat gets lost in the shuffle.
I use smaller drums and tune them a bit higher when I play live. I muffle very, very little - a rolled-up T-shirt in the bass drum, maybe a Zero Ring on my very live steel snare - and I tune for sustain and resonance. This gets me ringing and a little snare buzz when I'm alone. But then I slip on my isolation headphones and the drums sound far more like what I would hear out front - full, deep, throaty, and able to project through a wall of muddy midrange. Experience and live recordings have borne out my conclusions time and again.
To hear my drums on their own, I'm sure I would sound like your jazz guy, higher tuned drums, a little ringing, some overtones. And sure, when I go to record, I tune it a little lower sometimes, and I might slap some Zero Rings and Moongel here and there to tame things. But I also love the big sound a live room gives my drums, and often my preferred method is a single stereo dynamic mic about eight feet away, catching the room sound of my set, with few changes to the drums themselves.
I take a lot of clues from the pioneers and masters, who very instinctively knew how to get the sound they wanted out of any drum or cymbal, without digital compression or triggers or esoteric heads or drum dials. They worked on their technique and playing to make the drums sound sweet. Jason Bonham has told the story of his dad coming in and playing his little kid-sized drumset - and making it sound very very similar to his full-sized Bonzo tubs. There's something in the hands, the feet, and the soul that can't be got with a tuning key, IMO.
My final thought on this, and I speak as a multi-instrumentalist - tuning is crucial to even play a melodic instrument correctly, and it's one of the first things you learn to do as a guitar or horn player, for instance. But it's one of those things that many drummers don't know about their instrument at all. And sometimes I shrug my shoulders and idly wonder why that is.