Temperature is just an indication of a combined property of the masses of the molecules and their random motion. In principle, we can explain "no effective energy transfer between two conducting solid bodies in contact" via a condition in terms of the masses of the molecules and their speeds such that due to the collisions of molecules of two bodies, net energy transfer between two bodies is zero. But it would be a complex calculative work to derive this condition analytically so we use the temperature scale just as a phenomenological parameter to easily determine the condition of "no net energy transfer between conducting solids" for practical purposes. But it does not denote any fundamentally new property of the body separate from the already known mechanical properties of the same. Then why do we call it a fundamental quantity, e.g. in the SI list of fundamental quantities?
Answer
"Temperature is just an indication of a combined property of the masses of the molecules and their random motion."
No! Temperature is not always limited to being a combined property of the masses of the molecules and their motion. Of course, this was the first scenario where the notion of temperature became apparent to the humans historically, but our modern notion of temperature transcends this primitive notion of temperature as being some kind of a measure of the kinetic energy of molecules. Rather, the temperature is a quantity that generically represents whether a given system will be in an equilibrium when kept in contact with another system. More specifically, it represents whether the two systems can exchange energy with each other and attain a combined final state with more number of compatible microstates than the number of microstates compatible with the combined initial state of two systems. If they can then they would evolve to that state and otherwise not. In fact, we don't even postulate that such a quantity must exist but it follows from the basic postulates of statistical physics that such a quantity would exist and then we identify this statistical quantity with the thermodynamic temperature to make a contact between our theoretical framework and the experimental results---as needed to be done in case of any theoretical framework.
Now, the key is that this statistical quantity that we identify with the thermodynamic temperature is rather general and conforms to our primitive notion that ''the temperature has to do with kinetic energies of molecules'' only if the Hamiltonian of the system is that of a classical ideal gas, $H=\displaystyle\Sigma_{i}\dfrac{p_i^2}{2m}$. There certainly exists very many Hamiltonians (that is to say, very many systems) where there might be many other terms in the Hamiltonian which do not represent the kinetic energy of molecules and there might not even be any notion of the motion of molecules at all (e.g., there are no kinetic energy terms in the Hamiltonians representing magnets etc.--and yet, the concept of temperature as a quantity defined in the statistical sense we discussed makes perfect sense on its own!) So, in short, the text in the blockquote is not really true in the light of our modern understanding of temperature.
Now, as to whether the temperature is a fundamental quantity or not, as clearly evident from the statistical definition of temperature, the quantity called temperature emerges from the more basic statistical considerations and is not thus fundamental in the sense that it is not irreducible to more basic notions. But certainly, it is a very important quantity for both theoretical and experimental purposes and can be regarded as fundamental in this sense. Whether it absolutely requires a unit of its own has a definite negative answer. But again, from a theoretical point of view, every quantity can be expressed in terms of just one unit, say $eV$---but clearly, it wouldn't be convenient and thus, it is certainly wise to use a separate unit for temperature (and other quantities as well) despite the fact that we can use a more unified framework of units.
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