And yes, this person could make use of it. His videos are among the highest quality science explainers - he’s like the 3B1B of first principles in physics. Truly a savant at creating experiments that demonstrate fundamental phenomena. Seriously check out any of his videos. He made one that weighs an airplane overhead. His videos on speed of electricity and speed of motion and ohms law are fantastic.
But in principle, a LIDAR could be reconfigured for the purposes of such demonstration.
If one wants to build the circuit from scratch, then specifically for such applications there exist very inexpensive time-to-digital converter chips. For example, Texas Instruments TDC7200 costs just a few dollars and has time uncertainty of some tens of picoseconds.
I think some of the short range depth cameras (Kinect v2 was one) use time-of-flight technique, and could in principle be reconfigured to perform a similar demonstration, though it would not be as "homemade" and cool as the system built for the Youtube video.
Hard to justify going with the legacy brands as hobbyist with what comes out of China these days. Rigol and Sient make very capable hardware although h the UI can be a bit painful if you’re used to something else. Micsig produces very capable probes for a small fraction of the cost of the legacy players even high end specialty stuff like optically isolated probes. Those are still thousands but try pricing the equivalent from tek or Agilent, it would buy you a sedan.