Not sure about the healing thing.
A Fighter 5 with the standard 2-3 short rests per long rest is gaining an extra 3-4 1d10+5 Hp per day baked in via second wind, and that's in addition to potions of healing which are generally common enough to purchase. HP recharge to full on long rests, and he can expend HD on short rests too. Also: healing magic is readily available in 5/12 classes on offer from 1st or 2nd level.
Of all the in game resources, HP are the easiest to replenish.
Math time!!!
So a level 5 fighter with a 14 Con has about 44 HP. He also has 2 HD per day to spend, for another 16 HP. His daily HP is 60. He also has 10.5 HP from each use of second wind. Not bad!
Now a group of 4 PCs fights a Hill Giant and 2 ogres (a hard difficulty fight). The giant does about 23.5 DPR to a 17 AC PC and the Ogres deal about 7 each to a 17 AC PC. This means that combined, they deal about 37.5 DPR.
Now, a group is supposed to be able to get through 6 such encounters per day. Assuming they each of these battle lasts for only 3 rounds (a rather low estimate), that means the party takes roughly 112 damage per combat. Let us say the monsters are dumb and never focus fire, and spread that damage around evenly among the
party. That is 28 damage per party member every combat. Now with just 6 combats per day, that is 168 damage per party member.
So for our fighter to get through this much combat, he would have to take 10 short rests per day, using second wind at every possible opportunity. Of course, the other classes who lack second wind wouldn't be able to keep up. The party rogue, warlock, or Bard with 14 CON only have 38 HP and 14 HP from HD. The party wizard or sorcerer with 14 CON would only have 32 HP and 12 HP from HD. The fighter can't fight on his own, so it doesn't matter how much HP he can regain from second wind if the other party members have no HP left.
Even using every single spell for healing, a cleric, Druid, or bard are only providing about 140 HP per day to the party.
Also, yes my fight assumes the monsters are capable of fighting for all 3 rounds which is unrealistic, but it also assumes all PCs have a 17 AC at level 5, and that they can kill the monsters in only 3 rounds, and that the monsters spread damage around evenly instead of focus firing (which are equally unrealistic assumptions). Even if the monsters only had 2 rounds of up time, that still averages 112 HP damage per party member over the course of the day, which is 2.5x what most PCs can effectively take.
This is why the 6-8 encounters per day assumption is bogus. Players simply lack the endurance to take that kind of punishment. 3-5 encounters per day is far more realistic, especially if your DM ever throws encounters harder than medium at the party.