Honestly I'll go with any system that seems right for the game. For instance- you don't play unreal tournament 3 the way you play gears of war, and you don't play gears of war the way you play halo, and you don't play halo the way you play half life 2. Half Life 2 and UT3 use armor health so why don't they play the same? They're just different games, and the systems they have are appropriate to them. Basically what I'm saying is, if you don't like health regeneration, don't play the game that has it. The statement that health regeneration in games like Halo and GoW is 'dumbing down' the game for simpler audiences is phenomenally judgmental and rather elitist. The games both have plenty of challenge in my opinion. In point of fact, playing halo 3 and half life 2 on the same difficulty, I have a harder time on halo than half life.
Which brings us to checkpoints, what's the difference between checkpoints and lives other than their unlimited use? I can't really see one. As far as I can tell checkpoints are really a console thing, that since you own the space and equipment, allow you to say 'okay I'm done for the moment/and/or need to go do something else' without losing tons of progress, with computers you just save where you are, essentially they remove checkpoints so that you can make your own. But I don't hear people griping about being able to save their damn game and quickloading it.
I didn't play games like contra or bionic commando, so I'm not going to denigrate them, I did play Sonic and Mario, which featured lives and continues, and I did play some of the old 2d fps games and honestly those just weren't shiny for me, watching me on-screen die and pop back at a position maybe a few paces prior, sans whatever I had was less frustrating and more depressing. It didn't really make me feel better if I ever beat it, just made me wonder why I bothered.
Sense of satisfaction from the process, or sense of satisfaction from the conclusion. I prefer to enjoy things as I do them, so for me, it's whatever system best enhances the game.