I tried it with a regular 2” or so long needle from the sewing kit I use to sew buttons on with, and it threaded perfectly three times in a row without a problem. The eye in the needle was probably twice the diameter of the thread, whatever they typically are. Certainly not oversize.
It looks like the flat part of the needle has to be against the skin.
My German grandmother was a seamstress. 70 years ago she showed me how to thread needles just like this and make knots by rolling the thread between her fingers.