Re: margin of error
"How do you know that eclipse measurements from 2000 years ago were accurate to the second?"
They weren't. No-one has tried to look at historical data down to the second, they looked at the accumulated difference, which is much larger. Each day since 720BC has been on average 24ms longer than it would have been without any slowing, so we're now nearly 7 hours offset from where we would have been. It's exactly the same as having a clock that runs very slightly slow; you won't be able to see any difference if you put it next to an accurate clock and look at the second hands, but put the same two clocks next to each other a few months later and the slow one will show a completely different time. And yes, astronomers 2000 years ago were easily able to calculate eclipses with better accuracy than several hours, which is how they can tell how much the Earth has slowed down since then.
Note that this is also the case for leap seconds; they might sound big next to deviations of milliseconds, but again they correct the cumulative drift. There's a nice graph on Wiki that shows this - https://upload.wikimedia.org/wikipedia/commons/5/5b/Deviation_of_day_length_from_SI_day.svg ; just a millisecond or two difference in the length of a day results in 27 seconds difference in what a clock says the time actually is after a few decades of that small difference constantly adding up.
As for the result only being an average, that's true, but not especially relevant. The Earth is really, really big. Anything capable of making a significant step change in its rotation would be utterly catastrophic, and certainly no such thing has happened in the last couple of millennia. While the change has not been exactly a constant 1.8ms every century, it has certainly been a steady, somewhat meandering drift and not a sudden change at some point during the period studied. As that graph shows, there's a 1ms or so seasonal variation, a somewhat larger short term drift, and then the longer term drift which was the subject of this paper; even the largest earthquake on record couldn't cause a sudden 48ms step change.