My personal opinion is that the older style of gameplay where you have a health bar and no savepoints at all during the level is fine but the correct style completely depends on the game.
For example, half life 2 in my opinion is one of those games that could not work without the checkpoint system it has as the way the game is designed encourages you to experiment with things like the gravity gun and makeshift weaponry.
Something like cod4 on the other hand is well suited to its health regen thing as its meant to be a fast paced game and this enables you to play a lot more aggressively than you would with a health bar.
However, in my opinion there is only one game in recent years where the system used is absolutely perfect. Swat 4. They got the health and save system perfect in that game. No saves at all in the level, no health bar but different parts of your body that can be injured, for example, get shot in the leg and you are slower and have a noticable limp, take one to the arm and your aim is worse and it takes longer to move your gun and aim.
Anyway, the point of this ramble is that there is no perfect gameplay style, it depends 100% on what sort of game is being made and how the game is meant to be played.