Why is flux-linkage a fundamental circuit variable?
There were historically 5 fundamental circuit elements: resistor, capacitor,  inductor, and current and voltage sources. They could be described by very simple equations:

Voltage source: V = constant with respect to I

Current source: I = constant with respect to V

Resistor: \$V = I R\$

Capacitor: \$Q = V C\$

Inductor: \$\Phi = I L\$

with \$Q = \int{I dt}\$ and \$\Phi = \int{V dt}\$.

If we extend the concepts of these fundamental circuit elements to include nonlinear behavior, and multiport behavior (controlled sources, 2-port networks), we can model all other circuit elements we use. For example, a diode is a nonlinear resistor, and a transistor can be described by a nonlinear 2-port impedance matrix.

\$\Phi\$ is considered a fundamental circuit variable because it is the variable that describes the state of an inductor, just as stored charge defines the state of a capacitor.

The memristor, of course, is the missing "fundamental circuit element" that connects charge and flux linkage together without reference to voltage or current: \$\Phi = M Q\$ (someone correct me if I've inverted this one).