When Jesus does friendship evangelism, it doesn't look anything like what we call friendship evangelism. It starts, not ends, with a presentation of the truth. Genuine care for the unbeliever is shown in the context of truth, not in its absence. Unfortunately so much of what passes as friendship evangelism today is neither friendship nor evangelism. Unsaved people may be a lot of things--dead in their sins, enemies of God through wicked works--but they are not stupid. They know when we have an agenda with our so-called "non-confrontational" friendships. There is another word for people who use friendships for a secret agenda--"manipulation." No one likes to feel manipulated. When it comes from someone who is religious, it has even more of a stigma in this culture.
What if Christians were straightforward about their agenda and developed friendships based on truth and genuine care for the lost? What if we proclaimed the gospel on the front end of our relationships until waiting until we "earn the right"? What if our relationships had the savor of the grace of God TOGETHER with a bold proclamation of truth? Wouldn't that be powerful? Wouldn't that be what Jesus did?
One more thought on this and I am done. If you hold to sound, Reformed theology and believe that regeneration is an act of the Holy Spirit using the "incorruptible seed of the Word of God" (Jn. 3; 1 Pet. 1:23-25), then shouldn't your evangelism reflect that? You don't need to help the gospel by your relevance; you just need to let it loose. Helping the gospel by over-contextualizing or by being over-relevant makes as much sense as strapping a BB gun onto a nuclear warhead in order to help it out. The evangelism advocated here is simple and easy. Open your Bible, read it to a lost person and repeat.