The only other approach I've seen goes back to the basic assumptions in the trope I mentioned above. When evangelicals say “God wants a personal relationship with you” they aren't quoting Scripture but paraphrasing its ideas. The Bible speaks of God as our Father, and as our Husband or Lover, which certainly sound “personal”; but other possibilities suggest themselves to my uneducated mind. I wonder if historically these metaphors have been understood to refer to God's relationship to communities, not individual believers: the Jewish nation, e.g., or the universal Church.
At any rate, suffice to say there are understandings of the Christian faith which do not at all necessitate a direct personal relationship to God except perhaps as mediated through the Church. Mystical experiences are not necessary in these models, but neither are they discarded, just decoupled from the question of salvation.