Given the similarities between Lift and drag and their very similar relationships to the density, surface area, and airflow velocity, could one, in theory, describe aerodynamic lift as a kind of drag where most of the force from lift is just directed in the vertical direction where this vertical component of the force opposes gravity?
An 'aerodynamic force' (just one force...) appears when a body is immersed in a fluid stream. By convention, two components are chosen, one of them parallel to the stream direction, called 'drag', and the other one, perpendicular to that 'drag' is termed 'lift'.
At least as the terms are normally used, lift is not drag, and drag is not lift.
What I suspect you're looking for is: "induced drag", which is the part of the drag that's caused by an airfoil generating lift.
Induced drag is opposed to "parasitic drag", which includes things like friction drag (friction between the airfoil's skin and the air) and interference drag (e.g., there's some extra drag where the wing mounts to the fuselage) and so on.
Using the wing as a frame of reference, then the aerodynamic force is related to diversion (change in direction) of the relative flow. In a normal situation, the relative flow in front of the wing is horizontal, and the wing curves the flow downwards a bit. Assuming an ideal case where the flow speed isn't changed, then the horizontal component of the initial velocity is decreased somewhat (induced drag), and the vertical component is increased from zero to some non-zero value (lift). In a real world situation, the flow speed is reduced due to friction, viscosity, profile drag, ..., which further reduces the horizontal component of initial velocity (other components of drag).
Using the air as a frame of reference, drag is related to the forward acceleration of air, while lift is related to the vertical acceleration of air. Consider the case of a bus on a highway, the air is accelerated forwards, most of this due to the low pressure area behind the bus, some due to the air being pushed forward and around the front of the bus, while there is zero lift.
On a side note, lift calculations are simpler than drag calculations because lift calculations can be based on flow field calculations that calculate speeds (relative to the wing) and the coexistent pressures related to those speeds. Drag calculations are more complicated because they need to take turbulent and boundary layer effects into account.
Maybe the best answer here is that, while lift and drag are separate quantities, their behavior is governed by the same physics ie conservation of momentum and Newton’s Third Law of Motion.
One might ask what application would considering lift as a type of drag have? They are compared as L/D ratio as a parameter of design performance, however, especially as pilots, knowing the distinction may be extremely important as they apply to flight.
The amount of lift one can generate is dependent on velocity (for a given aircraft). Why this is critically important is that velocity is the kinetic energy state, which is created by adding energy to the system from the thrust. A little like filling a barrel. One spigot is altitude (potential energy) that you can pour back in as needed. The other is drag. Thrust keeps filling the barrel, but it must be filled to a certain minimum level or the plane does not have enough velocity to fly.
Empty the barrel too fast (steep climb), you can grab your potential energy (altitude) and pour it back in to get flying speed back. The thrust can only be added at a certain rate, as we know on our take off roll.
Drag finds application as how much energy is being taken out (as related to fuel consumption or max velocity) and can be compared with lift for a given configuration, AoA, airfoil type, etc.in engineering theory, but as far as piloting, not related IMHO.
I think that's the wrong way to look at it. The lift force is 90 degrees to the airstream. When you're gliding, the lift force is producing thrust, not drag, or else you would stop dead and fall from the sky.
I think of the lift-related drag component as the energy consumed in inducing the air to move from point A to point B, creating the pressure differentials and down thrust from action/reaction, as the wing passes through it.