In a sense to be discussed below, clocks do indeed measure time, and this is a very definite experimental result that gives us an experimental definition of time.
We experimentally observe that the ratio of the rates of the same two physical processes taking place in an inertial laboratory is always the same. A clock pendulum swings a set number of times, as a rubidium atom in an atomic clock oscillates a set (generally mich higher) number of times before a certain, always the same, extent of reaction between the same chemical reagents is reached, and this number of swings and the number of rubidium oscillations before that extent of chemical reaction is always the same if the pendulum, rubidium and the reagents are at rest relative to one another. It's part of the experimentally observed predicability of the World: set up the same two experiments with the same set of conditions and the physics will be repeatable: the ration between the rates of progression of the experiments will be the same as long as the two experiments are at rest relative to one another. It is this consistency between rates of processes that lets you pull an egg from boiling water when the sands of your egg timer have run out, and to know it will be cooked a consistent amount defined by your egg timer, even though there is no direct causal link whatsoever between the timer and the egg.
Given this basic consistency, the notion of a "good clock" becomes well defined. It is simply an instrument whose behavior is repeatable enough that its rate of working relative to the physical processes around it is always the same. Note that this notion would not be well defined if relative rates between the same physical processes weren't consistent and changed randomly. Misner, Thorne and Wheeler have a wonderful discussion of the notion of "good clock" in the first chapter of their book "Gravitation", as does Ben Crowell in the early part of his book "General Relativity".
We choose a "standard" cyclic process, measure its rate (or period), and then define the "duration" of all other processes and the "time elapsed" between pairs of events as the number of "standard cycles" that complete throughout the process or between the events concerned.
Likewise, when the same ratio is computed for the same pair of physical processes happenning in different inertial frames, the ratio of their rates changes from the value it had when they were relatively at rest, and this change of ratio is given by the effect on the time co-ordinates of each frame by the Lorentz transformation between the frames. This too is a strongly confirmed experimental result, even though we guessed the right transformation grounded on symmetry and other theoretical arguments some decades before it was confirmed by measurements.
It's all yet another manifestation of the experimental result that Eugene Wigner called the "Unreasonable Effectiveness of Mathematics in the Natural Sciences - processes can be foretold and the World isn't total chaos. There is a repeatability in physics.