If the horizon drop is millimetres then you would barely be able to notice it in your field of view, you could certainly be able to see it. Most telescopes and binoculars do not have a crosshair on them, only if they are used as a surveying tool, known as a theodolite.
But people with these devices have done the test you said and found:
There are many others, all showing the drop you predicted would happen if we lived on a globe.
Well done, you have successfully proved we live on a globe. Finally time to put this thread to rest and for you to have a quiet pint tonight to rest easy and realise they are not all out to get you, not all out to con and trick you, it is just the world doesn't always match our imaginations.