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."