I want to ask a fundamental question about Lorentz force equation. Why is it important to incorporate both electric and magnetic forces into one single expression? I know magnetism is electricity's compensation for relativity, but is it the reason behind?
Well, you could treat them separatedly, via two equations, say
but since Newton's second law holds, in presence of an electric field and a magnetic field, the total force will be the sum of both, that is, the Lorentz force.
I would say is just as simply as that.
Why is it important to incorporate both electric and magnetic forces into one single expression?
Because the Lorentz force is "due to electromagnetic fields". I know people talk about electric fields and magnetic fields, but see Wikipedia: "Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole - the electromagnetic field". When we only see linear "electric" force that makes an electron move in a straight line, we talk about an electric field. When we only see rotational "magnetic" force that makes an electron move round and round, we talk about a magnetic field. But these aren't really two different fields, they're merely two different aspects of the electromagnetic field. You know how people say that when you move a charged particle you create a magnetic field? It isn't true. You can work this out by moving relative to a charged particle. You don't create a magnetic field for that charged particle just by moving. Instead you now see an aspect of its electromagnetic field that you didn't see previously.
I know magnetism is electricity's compensation for relativity, but is it the reason behind?
Kind of. Minkowski spoke about electromagnetism in Space and Time:
"In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect".
Note how he said the field? It's one field and two forces. The electron doesn't have an electric field or a magnetic field, it has an electromagnetic field. Electromagnetic field interactions result in linear "electric" force and/or rotational "magnetic" force. The force-screw is something like a screw-thread: you push current up a wire and the motor turns, or you turn the dynamo to push current up a wire. But this isn't quite relativity because Maxwell referred to it in On Physical Lines of Force: "a motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw". To get a handle on the time-axis, see electromagnetic radiation on Wikipedia:
"E and B far-fields in free space, which as wave solutions depend primarily on these two Maxwell equations, are in-phase with each other. This is guaranteed since the generic wave solution is first order in both space and time, and the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time..."
One's the spatial derivative, the other's the time derivative. If it was a water wave and you were in a canoe, the tilt of your canoe is E and the rate of change of tilt is B. You can't have one without the other.