Descent's a great example — I recently played Overload, which has attritional health (you get health drops from enemies you kill, iirc). This made it very dangerous to take large damage spikes, so there was a lot of careful room-clearing using the ricochet weapons before you actually went in full blast.
With a regenerating health pool you can afford to take bigger risks. This allows you to 'get in the ****', as you say, without taking permanent damage you can't necessarily recover from. And your tactics of "poke out, do a little damage, and hide" of course don't work in Halo, because your enemies have regenerating shields too—unlike in a Half-Life or Call of Duty where you can score permanent damage with little risk.
(It's worth noting that in high level play, Doom 2016 does NOT reward you for getting in close, exactly because the amount of risk required to perform glory kills is too high: despite the designers' best intentions, you spend more time doing gauss cannon combos than you do ripping and tearing.)
What I think you're really talking about is a trend of reduced total health, like in a Call of Duty game where you'll die so rapidly that you can't afford to leave cover for more than a moment or two.
Halo also has a ton of exploration, because health isn't the only possible mechanical reward for exploring a level. There are overshields, which work exactly like the 'there's a boss fight coming' signals you mention. There are camo modules and, of course, hidden power weapons. Most importantly, there are new approaches to areas, often involving using the game's physics to get vehicles and players where they're not 'supposed' to be. This is why Halo tricking and stunting has been so robust and popular.
By contrast, Half-Life 2 has a dreadful semi-procedural health drop system where you will always find the health packs you need, strewn around like salad in conveniently excusable crates and closets. This makes level design very easy, because you can always count on the player having plenty of extra health—after all, your game will spawn it! You can pile that health up as high as you need to in order to meet larger challenges, like a Combine gunship.
By contrast, a Halo-style regenerating shield system makes level design MUCH sharper and more difficult—you know the player has a fixed resource pool and you can calibrate your encounters exactingly to strain that pool to its limit. Want Gordon to fight off a platoon of Combine soldiers? Give him enough health packs to do it. Want the Chief to fight off five high-ranking Elites in close quarters? There's no way, short of dropping an overshield, for you (the designer) to give the player more health. You can only give them more TOOLS — weapons, geometry, knowledge of the Elite AI.
Quicksaves are dreadful for flow and removing the option to quicksave every five steps was a huge boon for designers. **** savescumming!