The brief summary given in the minutes linked by the OP is a serious misrepresentation of what Arkani-Hamed really said. I could not imagine that a theoretical physicist of his caliber would ever say such things. I had to track down a
transcript of his presentation, which naturally says things that are much more reasonable.
First, to get this out of the way. Arkani-Hamed did not say that space and time are going to disappear. He said that our present concepts of them are inadequate and will need to be replaced. Nor does the summary linked by the OP say anything like that.
From the OP link.
What was really said has nothing to do with the mass of the accelerator and the effect of gravity on it. The limitation is that ultimately the energy density produced in an accelerator would be sufficient to create a black hole, preventing the usual shower of particles and energy that is examined for interesting stuff. This puts a limit on how deeply we can probe.
Again from the OP link.
This is simply totally different from anything Arkani-Hamed said. It has nothing to do with the size of the measuring device. There is a fundamental limit to the accuracy of observations due to the Heisenberg Uncertainty Principle. Repeated measurements are not going to get around that. You cannot simultaneously and accurately measure two complementary variables (OK that is really Bohr complementarity). You can make both measurements as precise as you want. That is, the needle on the gauge does not quiver to use a metaphor.) But on subsequent observations you will realize that the answer you expected is wrong.
Arkani-Hamed also delved into interesting subjects, such as vacuum energy, rolled up dimensions, the expansion of the universe and others, that are not represented in the brief summary in the OP linked minutes.