Is it an integral transform thing, like how spectrum analyzers can claim super low noise floors if you sort of gloss over the "noise is proportional to badwidth" part and look in a tiny bandwidth without normalizing?
We also use techniques called power and signal recycling to enhance this bandwidth-sensitivity tradeoff even more. Combined these techniques give you what remains between your 1/1000th wavelength and the actual sensitivity of LIGO and Virgo.
[1]: https://www.optica-opn.org/home/newsroom/2019/december/squee...
Like you suggest, and adding to what sleavey mentioned above, I would say the answer is: averaging over time and space. The laser beam is pretty wide, so it averages over a significant area of mirror surface. (The optical system also selects one spatial mode of the laser beam.) And the stated displacement sensitivity ("1/10000 the width of a proton") only occurs when you integrate over the sensitive frequency band.