The reason we get involved with reactance is to avoid dealing with differential equations. Reactance is easy, differential calculus is hard.

Reactance only applies to sinusoidal steady state. Transient behavior requires differential equations. Linear differential equations like those found in most circuits are not too bad.

I'll go with what most folks believe,

Good thing we don't live in Columbus's time, or you would believe the Earth is flat. Similarly, in Galileo's time. a lot of folks believed the Earth was the center of the universe.

because we can honor Georg Simon Ohm for his contribution by naming anything we want as Ohm's Law; not necessarily what he proposed in the 1820's.

Where is the honor in naming a definition a law? We define speed as distance/time. Ever heard of Newton's speed law? I haven't.

Back then nobody knew what galvanic electricity even was, not even Ohm who used Fourier's heat diffusion equations to explain his experimental results (because everyone back then recognized Fourier as a smart guy.)

It doesn't matter how the electrical energy was obtained. They had to know right away that certain materials conducted current better than others. Ohm noticed the linearity was a property of the material. If Fourier helped Ohm out, no problem. Scientists often make advances based on the work of others.

There is no reason to refer to some crackpot physicists when we can read any introductory textbook on Electric Circuits to learn everything anyone would ever need to know about Ohm's Law.

No, you can learn everything you need to know from a textbook about the definition of resistance or impedance. Even if they mistakenly call it Ohm's law.

Or possibly, it is not the author of the physics textbook who is the crackpot.

Until you read the textbook and refute it, you cannot legitimately say that. Until you do the foregoing, that is just a wild accusation.

For you edification, there are some materials that are ohmic and others that are non-ohmic. That is one of the properties of a material. This link explains what that property is and how to determine it.

http://physics.kuniv.edu.kw/phys107/Exp1.pdf
Current flows into one terminal, the same current flows out the other terminal.

That is TTT. Charge flows, current is already defined as charge flow. Current flow literally means "charge flow flow". The current into the terminal has the same value as the current leaving the opposite terminal. That is because the same amount of charge accumulating on the capacitor plate per unit time is the same as the amount of charge leaving on the opposite plate per unit time. That does not mean that current exists through the capacitor. If it did, the capacitor would be a resistor.

Current flows through whatever is between those terminals.

No it doesn't. There are two circuits involved. One circuit accumulates the charge, the other depletes the charge. The two circuits are isolated from each other by the capacitor's dielectric. The charges that enter a capacitor are not the same charges that leave.

And the capacitor obeys Kirchoff's current and voltage laws just like any resistor must.

No it doesn't. Circuitwise yes, but on the capacitor itself, no. Kirchoff's law does not apply to a capacitor, because a capacitor stores and dispenses charges.

If there is a capacitor in a branch of a circuit network, will mesh current flow through the capacitor?

No, for the reason already given.

How would it be a useful concept for circuit analysis if current did not flow through a capacitor? Or do you propose changing the notation for mesh currents when they reach the capacitor terminals to show that the mesh current does not flow through the capacitor? How would that be more useful?

Mesh analysis works because the capacitor stores and dispenses charge by equal amounts per unit time. This fools a lot of folks into thinking that the current exists through the capacitor. The mathematics of mesh analysis does not know the difference, but those who understand how a capacitor works know what is really happening.

Ratch