As far as I understand, Einstein’s original formulation of Special Relativity did not include the Clock Hypothesis, though it was implicitly assumed.
The modern formulation of SR adds a formal statement of the Clock Hypothesis to Einstein’s original set of two postulates.
When was the need to formalize the Clock Hypothesis identified and when did a three postulate presentation of SR first appear in a standard text?
Introduction of the clock hypothesis is usually associated with Rindler's Special Relativity (1960), and the current debates over it were sparked after Brown defended its necessity in his book Physical Relativity: Spacetime structure from a dynamical perspective (2005). According to Arthur's chapter Minkowski’s Proper Time and the Status of the Clock Hypothesis, one can trace the hypothesis to a remark Einstein made to Sommerfeld (published in 1913):
"In passing I note that, in their insistence that the clock hypothesis is a separate assumption in SR, Rindler and Brown could appeal to the authority of Einstein. For in the notes he made on Minkowski’s original “Raum und Zeit” paper, Arnold Sommerfeld attributes a remark to Einstein that may indicate the origin of the idea that the clock hypothesis is needed – at any rate it is the first statement of it that I have been able to find. Right after mentioning Minkowski’s remark that $d\tau$ is not a complete differential, and noting that this shows that the proper times of two motions connecting two world points will generally differ (as discussed in Sect. 2 above), Sommerfeld adds:
"This assertion is based, as Einstein has stressed, on the (unprovable) assumption that the moving clock actually indicates the proper time, i.e. that at each instant it gives the time that corresponds to the instantaneous state of velocity, regarded as constant. The moving clock must naturally have been moved with acceleration (with changes of velocity or direction) in order to be compared with the stationary clock at the world-point $P$.""
Arthur and others (see e.g. Valente, What do light clocks say to us regarding the so-called clock hypothesis?) do not consider the clock hypothesis to be on a par with the standard two postulates of special relativity, but classify it with what philosophers call coordinating principles. Those relate theoretical notions to measured quantities, proper time to clock readings in this case, and are not part of the theory proper. They are needed for any theory to make it testable and not just a mathematical abstraction. Here is Arthur's summary of the positions:
"Rindler claims it is an assumption that it is necessary to make in order to get from “purely kinematic laws about acceleration” to the dynamics of really accelerated systems (1966, 28) and Harvey Brown claims something similar in his recent book (Brown 2005, 9)... On the other hand, Jim Hartle holds that Minkowski’s formula for the proper time holds “even for accelerating clocks, i.e., when the velocity is dependent on the time” (Hartle 2003, 62), and he makes no use of the clock hypothesis in his textbook. Roberto Torretti allows that the clock hypothesis “may be viewed as a conventional definition of what we mean by clock accuracy, and hence by physical time” (1996, 96), but argues that “Special Relativity would doubtless have been rejected or, at any rate, deeply modified, if the clock hypothesis were not fulfilled – to a satisfactory approximation – by the timepieces actually used in physical laboratories”...
On these latter views, provided a given process approximates well enough an ideal clock, the clock hypothesis seems to amount to little more than the desideratum that, with the metric locally Minkowskian, the predictions of SR should agree with experimental fact. So the question is, why should it be necessary to state it as an independent hypothesis? Two main sets of considerations have been adduced. As we have seen, one kind of justification has been that, since many natural clocks are subject to accelerations which result in their failing to satisfy the clock hypothesis, we need to appeal to the hypothesis in passing from the kinematics of acceleration of ideal clocks to the dynamics of really moving clocks. A second kind of justification has to do with the different status of Special Relativity within the General Relativistic context, where, as is shown by the example of Weyl’s unified theory of electromagnetism and gravitation, the metric could be locally Minkowskian and yet the rate of clocks could be path-dependent in such a way that the instantaneous rate would depend on the way the clock had been accelerated hitherto, contrary to the clock hypothesis. This, it is argued, proves the independence of the clock hypothesis from the assumption that spacetime is locally Minkowskian."
