As John Rennie mentions, the light clock relies on a precise measurement of the distance between the two mirrors, and that distance has to remain stable while the clock is running. It's much easier to measure time precisely than it is to measure distance, so we'd use an atomic clock to measure & maintain the mirror distance.

Also, we somehow need to precisely measure the moments when the light bounces off the mirrors, and the precision of that measurement is limited by the wavelength of the light; the shortest visible wavelength is around 400 nm, which corresponds to ~$1.33×10^{-15}$ seconds. We could use ultraviolet light for a shorter wavelength, and perhaps even x-rays, although you need to use very shallow reflection angles with x-rays.

It's simply not practically possible to make an Einstein / Langevin-style light clock that could have anywhere near the precision of a good atomic clock. The mirrors need to be positioned precisely, with no outside vibration. Of course, the bouncing photons will cause the mirrors to vibrate, presumably that's how we detect the reflections. The mirror temperature has to be kept constant, and the space between them has to be an ultrahard vacuum.

According to current international standards the second is defined by the Cesium Standard which is the basis of really good atomic clocks.

The caesium clock is good, but it's not the best timekeeper that we have. The first atomic clock was an ammonia maser built in 1949, but it was less accurate than the best quartz clocks available then. The caesium-133 clock was developed a few years later, in 1955. There have been some improvements in the design since then. The modern caesium fountain uses a small "cloud" of laser cooled atoms in freefall. Using the optical molasses technique, the atom temperature is reduced to ~40 microkelvin. However, even using a small population of such cold atoms, some thermal noise and atomic collisions are inevitable when the cloud is energised by microwaves. But even if we could use a single atom at picokelvin temperatures there would still be some width of the transition frequency, it cannot be a single number, due to time-energy uncertainty.

There's another issue lurking in your question. Time is what a clock measures. An SI second is defined as the time taken for 9192631770 oscillations at the frequency of the caesium-133 hyperfine line at 0 K. So it doesn't actually matter that each of those oscillations have a slightly different duration. You count 9192631770 of them and you have an official SI second. That is, the variation of the Cs-133 hyperfine transition is baked into the definition of the SI second.

We now have various clocks that have more precise ticks than the caesium clock. Such clocks can be used to calibrate a caesium clock, but to use such a clock as a primary time reference it has to reproduce the behaviour of the Cs-133 SI second, warts and all.

Here's a summary of the relative uncertainty of various atomic clocks, from Wikipedia.

"Beam" refers to a standard off-the-shelf beam maser. "Fountain" is an atomic fountain, that value is for NIST-F2. "Lattice" is an optical lattice. "Fermi gas" is a 3D quantum gas optical lattice.

NIST-F1 (also an atomic fountain) has an uncertainty around 5e-16. Together, NIST-F1 and NIST-F2 form the primary time & frequency reference for the USA.

FWIW, the SI second was defined in terms of an integral number of Cs-133 hyperfine transitions which approximately matched the current definition of the astronomical ephemeris second. The original ephemeris second was defined as 1/86400 of a mean solar day, but due to the variations in the solar day length & the practical difficulties in measuring it, the ephemeris second was redefined in terms of the mean tropical year of 1900, and prior to the adoption of the Cs-133 standard, ephemeris time was determined from observations of the Moon.

However, the mean solar second length used in those definition was calculated using data gathered from 1750 to 1892, and corresponds to the mean solar second from around the middle of that period, i.e., ~1820. The actual mean solar second is somewhat longer now.