I tend to agree with someone upthread (maybe [MENTION=6787650]emdw45[/MENTION]?) that this is evidence that the fights weren't equal.
For instance, in 4e a notionally "balanced" encounter is one standard monster of the PCs' level per PC - but I think that the typical PC is at least as robust as an elite of his/her level, maybe a bit more (especially at higher levels), for the sorts of reasons that you give (depth of offensive ability, but even more so depth of recovery abilities).
I agree with this for 4E. Almost every single fight, some PCs were bloodied and possibly unconscious and the foes were mostly ok. But with Healing Surges and healing spells, PCs became non-bloodied and attrition wins the day. Action Economy eventually shifts in favor of the PCs. It starts out against them because foes tend to use their encounter and daily powers immediately whereas PCs tend to start out with encounter powers (and only pull out Dailies if the encounter appears to warrant it).
In 5E, the typical story is different. In moderate fights, the PCs tend to have better AC, more spells/abilities, and similar hit points. So action economy shifts into their favor practically from the start of the fight. This is why healing is rarely used in combat except against tough or harder encounters. The fight is often over in 2 to 5 rounds (difficulty dependent) and PCs are not unconscious too often in hard or easier fights (although it can happen).
In order to get a near equal fight in 5E, the ACs have to be similar, the hit points have to be similar, the number of foes has to be similar, the damage has to be similar, and the to hit bonus has to be similar. If any of these are subpar by the NPCs, the DM has to throw higher level foes at the PCs to get near equality. With this type of nearly equal fight, now it is the recovery ability of the PCs that gives them the edge (and as you say, depth of offensive ability). They can still heal themselves. The foes typically cannot. When a PC goes unconscious, they can be brought back into a fight and shift action economy towards the group whereas the NPCs typically do not have this option.
And this is why Healing Word is generally considered superior to Cure Wounds. Cure Wounds uses an action this round to gain one additional action next round (assuming the healed PC does not go unconscious again). Action economy for this round actually shifts towards the NPCs. Healing Word only uses a bonus action, so action economy for this round does not shift towards the NPCs.