Part of that is just missing the era and the right nationality. Olga or Romani probably would have jumped to Hadrian's Wall immediately, and Da Vinci might have too if she hadn't been primed with, "Well, maybe it's Shi Huang Di," but Taylor is an American who missed way too much during her school years, the twins are Japanese, Mash is a clueless shut-in, Aife and Boudica are both too early to take it for granted, Arash is too early and too far away from it culturally, Spartacus is Spartacus, and the only parts of British history Emiya really cared about while he was alive were the King Arthur parts.
The way I looked at it was, "The only reason I know about Hadrian's Wall is because I watched that 2004 King Arthur movie with Clive Owen and Keira Knightley," and, "If someone were thinking about a powerful legend famous for building a continent-spanning stone wall, who would they assume it was first?" Since China's "Great Wall" is the most famous wall in the world, I figured Shi Huang Di is who they'd jump to first.
Whether or not the team is right in their initial assumption or really did miss the obvious answer, well, that's for later. All will be revealed in due time. I will say that it would probably make a lot more sense for Jing Ke to be there if Shi Huang Di was, too, but putting their confrontation so early would probably rob the one in LB3 of a lot of its weight. Make of that what you will, because I have said before that I will also lie to you guys sometimes when it comes to spoilery stuff for the sake of preserving the surprise later.