Could we not use these timebases as a planet-sized gravitational wave antenna array?

Maybe.

https://www.arxiv-vanity.com/papers/1501.00996/

https://ui.adsabs.harvard.edu/abs/2021nova.pres.8631W/abstra...

I don’t think the sample rate is high enough to detect individual waves.

Why would the sample rate be below 10 kHz?

These clocks depend dilute gasses of atoms. Usually the process for preparing and probing the atoms takes at least a few hundred milliseconds, and usually in the seconds range. There's a bunch of reasons for this, probably two main reasons IMO:

1. Generally for precision clocks you need "cold" or "ultracold" atoms (essentially random doppler shifts of hot atoms kill your accuracy). This means something like microkelvins - this is hard to prepare quickly. And atoms are constantly lost/heated due to collisions with stray gas molecules, even though the pressure is usually something like a quadrillion times lower than atmosphere. There's research into continuous ultracold atom sources, and you could prepare way more than you need and siphon a few off at a time, but both are technically challenging.

2. The way these clocks work is essentially you have some laser, and an atom that only reacts to laser light of a very specific frequency. If the laser frequency is off from the exact frequency you can tell. A variety of effects means that the measurement process takes some time. It's like trying to accurately measure your heart rate in 1s vs 10s - a lot easier to do in the latter case.

Nothing fundamental, just technically quite painful.

Of course it's some four-page Avi Loeb homework problem, er, I mean, "paper".