Damage, however, typically scales directly with the size/power of the weapon used.
There's a disconnect here. At first level, I can maybe take a hit or two from an ax... but probably not more than that. At tenth level, I may be able to survive a dozen such hits. Why? The answer is, typically, that hit points are an abstraction... that not all of those hits represent actual connections between the weapon and my body... rather they represent me wearing down my reserves and such.
If that's the case, then why does damage scale according to the weapon? Is fighting someone who is armed with an ax more strenuous than fighting someone who is armed with a dagger? It might be, but that certainly isn't how weapon damage is scaled.
Moreover, most people don't describe combat that way. In most games that I play in, if I hit on the attack, the damage gets described as a wound. Why? Because I hit on the attack. Describing the effect of a successful hit as a wound is natural. Similarly, healing is nearly always described in terms of healing wounds. This is behind the problem that many people have with healing surges in 4e.
One way to describe the problem here is to say that it isn't hit points and damage that should be considered abstract, but rather hits - most successful attacks shouldn't actually hit the opponent.
Once this is accepted, a new way to abstract combat opens up. Hit points represent physical durability. Weapon damage represents the deadliness of the weapon. A single actual hit from a dagger in the back might be able to kill even a high level character. The trick is that it is exceedingly unlikely that a high level character would ever get hit in the back with a dagger. We need to introduce something like abstract hit points and apply it to attacks. Borrowing a bit from FATE, let's call this Stress.
As characters advance in level, they don't really gain hit points. They might become harder to hit outright. They also gain more capacity for stress. What is stress? Well, when an attack would hit a character, that character can choose to take on some stress to prevent the attack from being successful. This might represent a near miss that throws the character off, but it could just as easily represent a twisted ankle or strained back that results from dodging (or falling) out of the way. All the abstractions that went into hit points are applicable here. When a character fills their capacity for stress, they can't take on more to avoid hits.
Stress and other subsystems
In a system with stress, armor should provide direct resistance to damage, whether that is a flat number or a variable. This provides characters with a meaningful choice of whether or not to take stress. If the threatened damage is likely to be absorbed by a character's armor, then taking on stress may be unnecessary (particularly if you expect to need that capacity later).
Healing could focus on primarily on hit points, but something like 4e's healing surges could be used to relieve stress. Once abstracted stress is separated from hit points, it makes sense for things like inspirational speeches to have an effect.
Various class abilities could trigger at specific stress points. Perhaps having stress makes it more difficult to cast spells. Warrior-types might gain the ability to use special attacks when they have a certain amount of stress (Final Fantasy Limit-Break-Style). There are several possibilities.
Perhaps (without certain special abilities), you can't take stress to avoid a surprise attack. This would make a dagger in the back a very serious danger for high level characters even without depending upon a backstab damage multiplier.