The problem is in the extreme weakness of the gravitational interaction compared to electromagnetism (and even compared to neutrino interactions), not the uncertainty principle. You can use small magnets to hang big things on a fridge door, even though the magnet and the object are being pulled floorwards by the gravitation of the entire planet. You need ENORMOUS vats of super-pure water surrounded by sensitive photodetectors <https://neutrinos.propg.ufabc.edu.br/wp-content/uploads/2019...> (Super-Kamioande, Japan) to occasionally spot light a particle strongly bounced by a neutrino from a significant nearby neutrino-generator. There are bazillions of neutrinos streaming through the chamber in that maintenance photograph. But there are A LOT more gravitons passing through, and that detector will almost never certainly spot an electron being sped up by them!
The idea in the paper is to replace water with something clever that is purer, much closer to absolute zero, and much more voluminous than Super-K. The clever bit is that it doesn't have to be more voluminous (by a lot) than our entire planet because interactions within the medium can (the authors argue) in principle magnify the signal of a deflection by a very precisely characterized (notably in frequency) graviton.
A somewhat unkind way of analogizing it is that they hope to see an avalanche and attribute it to (only) the landing of a very specifically shaped snowflake during a blizzard.