Phase-locked loops[0] are not magic ;)

Also I imagine they are using a bog standard Kalman filter[1].

0. https://en.wikipedia.org/wiki/Phase-locked_loop

1. https://en.wikipedia.org/wiki/Kalman_filter

I've implemented this (using GPS PPS to discipline a clock) using regular old PI control loops. I just make the loop damped enough to filter out the GPS PPS jitter and you're good.

From what I remember of textbook discussions of atomic clocks, stabilizing drifting oscillators is also in part how normal, old (50+ years) atomic clocks work as well, isn’t it? You have a fiddly and unreasonably high-frequency reference, you slave a relatively garden-variety standard like an environmentally controlled piezoelectric crystal to it, and then feed your periodic signals, counters, etc. off that.

Yes. Surprisingly, atomic clocks have high jitter compared to ovenized quartz oscillators. Every atomic clock package is outputting its signal from a quartz oscillator that is disciplined by the atomic oscillator.