On the basis of sharing the same Old Testament, I'd argue that we do. Given Jesus is said to have taught there and that is where he died for our sins, I'd say it very much holds a place in the hearts of Christians.
(You have no interest in visiting Calvary? Seriously?)
It's perhaps not as important as Bethlehem or Nazareth, but to say Christians should have no interest in it is a surprisingly odd answer from somebody with your interest in the religion.
I'd certainly like to be able to visit all three places without risk of harm or having to fund what amounts to a terrorist regime in sheep's clothing.
I think you misunderstand.
The city itself has no intrinsic value to it for the purposes of worship. Nor do Christians need to go there or somehow gravitate there at any time in their life.
Muslims and Jews, however, need the site itself. It holds value to their actual worship. They need access to it for their religious worship. Ergo, they fight for it.
I'd love to visit and I'd love to wander its streets and soak up the history. As a Christian tourist, it would be very moving.
But as anything else - it holds no value for my worship of God and it is not necessary to have anything to do with it. Jesus is the Temple, and the Spirit is our accrss to God. As such, from a purely human point of view, Christians should steer clear and avoid it unless giving humanitarian aid to those there.