Temperature: Why a Fundamental Quantity?

As I already commented one can introduce the Temperature of a gas by relatively modest assumptions. Here is a sketch of a derivation I hope to remember correctly:
The definition of temperature is then based on the concept that if two gases are brought together the entropy will maximize. This condition can be simplified to the condition, that the two inverse "temperatures" have to be the same. This yields the formula I already gave, namely $$\frac{1}{k_B T}=\beta= \frac{\partial \ln (\Omega)}{ \partial E}.$$ Here, $k_B$ is a scaling constant,$E$ is the energy and $\Omega$ something like the number of available states for the system with a given energy.
For a proper derivation you can have a look in practically every book on statistical physics.

It is one of the fundamental questions in classical thermodynamics.

Temperature: Temperature is the parameter that tells us the most probable distribution of populations of molecules over the available states of a system at equilibrium.

We know from Boltzmann distribution:

$$\beta=\frac{1}{k_B T} $$

The fact is that $\beta$ is a more natural parameter for expressing temperature than T itself.

Absolute zero of temperature (T = 0) is unattainable in a finite number of steps, which may be puzzling, it is far less surprising that an infinite value of (the value of $\beta$ ‚ when T = 0) is unattainable in a finite number of steps. However, although $\beta$ is the more natural way of expressing temperatures, it is ill-suited to everyday use.

The existence and value of the fundamental constant $k_B$ is simply a consequence of our insisting on using a conventional scale of temperature rather than the truly fundamental scale based on $\beta$. The Fahrenheit, Celsius, and Kelvin scales are misguided: the reciprocal of temperature, essentially $\beta$, is more meaningful, more natural, as a measure of temperature. There is no hope, though, that it will ever be accepted, for history and the potency of simple numbers, like 0 and 100, and even 32 and 212, are too deeply embedded in our culture, and just too convenient for everyday use.

Although Boltzmann’s constant $k_B$ is commonly listed as a fundamental constant, it is actually only a recovery from a historical mistake. If Ludwig Boltzmann had done his work before Fahrenheit and Celsius had done theirs, then it would have been seen that ‚ was the natural measure of temperature, and we might have become used to expressing temperatures in the units of inverse joules with warmer systems at low values of ‚ and cooler systems at high values. However, conventions had become established, with warmer systems at higher temperatures than cooler systems, and k was introduced, through $\beta=\frac{1}{k_B T} $, to align the natural scale of temperature based on ‚ to the conventional and deeply ingrained one based on T. Thus, Boltzmann’s constant is nothing but a conversion factor between a well-established conventional scale and the one that, with hindsight, society might have adopted. Had it adopted ‚ as its measure of temperature, Boltzmann’s constant would not have been necessary.

Conclusion: Temperature, actually is NOT a fundamental quantity. It is just for convenience and historical reasons we consider it as fundamental quantity.

Reference: Peter Atkins -The Laws of Thermodynamics: A Very Short Introduction

So why do we say it a fundamental quantity?

You do not have to say such thing, but temperature is very basic and important concept. In thermodynamics, it is the only quantity that always gets equalized in transition to thermodynamic equilibrium - pressure nor chemical potential needs to equalize, but temperature has to (except perhaps for systems in strong gravitational field, where the lower parts are predicted to have higher temperature than the upper parts).