5e doesn't do that either & it's a nontrivial change to make retroactive as wotc themselves show in the dmg. Linking the attrition to healing surges instead of hitpoints just shifts the focus of where the attrition is. 5e by comparison has a variant rule for healing urges that only works as anything but a huge power bump if applied to "parties that have no or few characters with healing magic", No mention on how to finish the variant rule so it works with parties with bard druid cleric some sorlocks etc because it's an extremely nontrivial thing to retroactively bolt onto 5e
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HS was just basically the 4e version of what 5e calls "hit dice". There was an action 'Second Wind' which a PC could use in combat, taking a standard action, and giving them healing worth one HS. This was not really a good move, you are better off attacking in 99.9% of all cases. Dwarves get it as a minor action, which is AWESOME for them, as they don't give up their attacks. This works fine though.
The whole model is quite elegant. You have HS which are your daily healing resource, and they are depleted by ALL healing, including healing word, second wind, healing potions, etc. CLW actually doesn't require an HS, and as a standard action power is not really a great option, though it is rather more useful if you have a healing focused cleric build. Short rests are 5 minutes and allow you to expend as many HS as you want, each heals 1/4 of your total hit points. So, GENERALLY you go into each fight with full hit points, or close to it. Leaders normally have 2 minor action heals (healing word for the cleric, etc.). Most parties have one leader, maybe certain other PCs may be present who have some healing, paladins for example. The point is, you are NOT guaranteed by any means to get 'popped back up', though probably that will happen the first time someone goes down. Also simply soaking up lots of damage is not a strategy, because you will run out of HS! While long rests recover all HS, there are obvious plot implications to an 8 hour pause. GMs also often rule that harsh environments inhibit recovery, partially or fully (I think this is also called out in various products, like DS).
The point is, while PCs ARE very resilient, they are far from unkillable, and while some PCs are likely to 'pop back up' at times, this is also less likely to be feasible in big fights (which tend to happen later in the day) and there is certainly a hard limit to this before you start giving up action economy to cast standard action healing. There are definitely tactics which can help with this, but frankly most good 4e players will tell you that a good offense is the best defense and wasting a lot of resources on masses of bulk healing and such are sub-optimal. A paragon Dark Pact Warlock is going to be a lot more help to a party than a paragon Pacifist Cleric, who has basically no damage output and simply stretches out each combat longer without having any good way to end them decisively.
The resulting system has lots of drama and tension around overall survival and success, and the GM can make things brutally tough (or not) as she sees fit.