How To Witness to Unbelievers: The Apostle Paul's Apologetic

Most of the time you will hear a Christian witness or evangelize an unbeliever by saying something like "Jesus died for you", or "Jesus has a wonderful plan for your life", or "If you died today are you certain where you will spend eternity?"
Is this how we are supposed to witness?  Is this how we see the apostles doing it in the Bible?
Perhaps of all places in the Bible, Acts 17:16-34 is the most relevant example to our modern, agnostic world.  In the account, we see Paul witnessing to Greek unbelievers.  Notice how he uses what they already believe or acknowledge.
Paul in verse 22 acknowledges their religiosity. In verse 23 Paul acknowledges their desire to "know" God. In verse 24 Paul points out that God can be knowable and that God is sovereign (see Romans 1:20). In verse 25 Paul further expresses God's sovereignty. In verse 26 Paul connects all men, showing that no man can claim a natural special status. In verse 27 Paul appeals to mankind's natural inclination to "know" God, even though without direct revelation they only grope around (see Romans 1:20). In verse 28 Paul EVEN APPEALS TO PAGAN SOURCES. In verse 29 Paul contrasts a living God with mere dead objects/depictions. In verse 30 Paul demonstrates God's long-suffering toward humanity. In verse 31 we finally get to something that we might say is specifically and uniquely "biblical", whereas from verse 22-30, Paul was appealing more to their innate awareness of deity. And what happens? In verse 32, they laugh and mock Paul and in verse 33 Paul doesn't hang around trying to push his point. He departs.  So often, we think if we could just deliver a better argument we can "prove" someone into believing (see 2 Peter 1:16).

In verse 34, we see some were moved by the truth and DID believe. All without Paul ever uttering, "Jesus died for you and has a wonderful plan for your life". We have to suspect these new followers were further discipled. Ultimately, we affirm, and Paul affirms by his apologetic that "no man comes to the Son unless granted by the Father" (John 6:65) And that flesh and blood does not reveal the Son, but the Father in Heaven (Mt 16:17)

Of course we are to quote Scripture to unbelievers, but people who claim we are to only quote Scripture, don't seem to understand Paul's apologetic. Scripture is for the initiate. It means nothing to people who's eyes are not opened, yet we can't know who's eyes have been opened until they respond. I am NOT appealing to "empiricism" or vain philosopy. I am appealing to Paul's own apologetic.