The problem with a mechanical clock isn't that it's "inaccurate" relative to some other clock. The problem is that it does not measure time at the same rate: a mechanical definition of a second drifts a lot with temperature and component aging, no matter how you'd implement it, and is also changing with gravitational acceleration and with Earth's magnetic field, unless you make the entire mechanical clock from insulators. Once you get to precise time measurements, even second- or third-order effects become significant.
Both Earth's magnetic field strength and local gravitational acceleration locally drift at rates that we can now currently measure quite well, and we can do that because we can measure time without a clock affected by such effects.
Now you ask: but why we care, if we just broadcast this changing second?
Ahem. F = m*a. If the definition of the second drifts around, so would have the definition of a kilogram, in order to get accurate results from basic things like Newton's 3rd law. 3rd law is indirectly used in all kinds of measurements, so a drifting second standard would be bad news. And that's just an elementary example. All basic physical constants are inter-related to the definition of a second...
And the solution we found was finding more stable mechanical systems. It just so happens that the more stable a clock's rate, the more reproducible the design becomes, as well.
And the atomic clocks are... mechanical clocks. Quantum mechanical, but still. It's comparatively simple to replicate them everywhere you are, just given their description, since their definition is tied to... wait for it... the definition of natural numbers and counting. As long as you can agree with someone about how you count the contents of atomic nuclei, you can agree about what elements to use for the atomic clock. You can similarly count the orbitals and zero-in on the state transitions used as the time base in the atomic clock. This gets more into the Contact (the book/movie) and the way it presented the concepts of communicating basic science across civilizational divides. You have to start somewhere, and natural numbers work quite well for that purpose.
Another important thing is that time is the quantity (of sorts) we can measure most accurately. It thus helps if we can tie, at the fundamental level, the definitions of other physical constants and quantities to the definition of a second. For example, the Josephson effect ties time (frequency) to voltage, and we suddenly could improve the accuracy of our Volt standard by order of magnitude compared to previous standards of e.g. electrochemical or themoelectric nature. This circles back to Newtons, since we can relate force measurements to electromagnetic force when a certain current flows through conductors, and we can define current in terms of passage of time and natural numbers (counting the electrons!).
So, in practice it turns out that having a highly locally reproducible standard of time can be used to propagate, or disseminate, other physical standards, since they don't need to be based on any broadcasts other than static shared knowledge and definitions. That's important, as e.g. how much you pay for electricity is tied to the definitions of Volt and Ampere, and if you want to meaningfully compare electricity prices between two countries, they better agreed on how a Volt and Ampere relate to other physical constants. And such agreement is best when anyone can locally derive the needed standards without asking anyone else for anything but information.
In a nutshell, relating other physical constants to a definition of a second that is quantitatively tied to the natural numbers and fundamental physical processes allows anyone and everyone to synthesize their own physical unit and constant standards independently, without any dynamic broadcasts or sharing of artifacts.