I understand how there could be a perfunctory religious practice without any life in it that would cause one to hunger and yearn for something much more personal. And the offer of relationship touches that hunger.
But here’s the problem: “having a relationship with God” in this sense is not at all unique to Christianity. Virtually every religion, it seems to me, has as its goal something intensely personal.
And this is why this distinction actually weakens our message if we’re not careful. It’s a version of “try Him you’ll like Him.” People can easily say, “No thanks, I’m trying something else right now and I like that just fine.” In fact, that’s been the response I’ve frequently gotten when I’ve offered Christianity to a non-believer with the appeal that it’s not religion but a relationship. Their response: “I’ve already got a relationship with God, thank you.”
The core of the Gospel isn’t having a relationship with God, but being rightly related to God.