I think it is just the draw of the word 'Everest'. I heard that you can't even see Everest from much of Everest base trek. Perhaps someone who has been can verify that?

It's true. Usually, the first time you see Everest is from the Everest View Hotel (a small hike up from Namche Bazaar) -- so around day 3, depending on your speed -- and then once or twice more on the approach to Gorakshep.

From there, you either hike up the Kala Patthar view-point which has that famous panoramic view of Everest, basecamp, and all the surrounding peaks... Or you trek around the canyon and begin the approach to EBC itself.

You can see it, but best views are from nearby hill called Kala Pathar. I presume you meant from Everest Base Camp, from EBC hike you can see Everest pyramid many times.

There is a very nice variant of EBC hike called 3 passes trek. Goes over 3 high altitude passes (5500, 5400, 5300m) on top of base camp, making a nice loop. A better challenge, once outside main valley just few people, and views are stunning, ie from Gokyo Ri, or north walls of Taboche or Cholatse. A difference between meh and stunning for me.