It does actually, plenty of times. Not only did Jesus state that no one has access to the Father except through him, he also says this is the case for him as well. No one gets to Jesus Christ except those the Father brought to him. In order for anyone to choose him, he has to enable us to be able to choose him first to start with.
Your ability to actually believe in God comes from God's will, not our will, and Jesus made that very clear, so did Paul, and I'm sure we all know the scriptures pertaining to the carnal mind. John 3:16 may be simple and to the point, but sometimes the simple answer isn't good enough. So the foreknowing comes before the predestination, or the method as to how one is saved. Is it Gods wish that any perish? No, but I really don't see how your explanation is any different than Calvinism when it comes down to it, since who gets saved is still based primarily on his omniscience, and not on his will. No matter how you want to spin that, the fact is God does the choosing, whether that choosing is according to "predestination" as you define predestination or according to what he knows doesn't really change much. God is sovereign, and he will have mercy and compassion for whoever he chooses to. This doesn't mean those he refuses to show mercy for is automatically doomed to the Lake of Fire, which brings me to your last question.
I'll rephrase this; What kind of God would choose some to the first Resurrection unto salvation and others to the second Resurrection of judgement? You see what I did there? I didn't say God chooses some to hell. Whose name gets written in the book of life is entirely up to him, and it will be primarily according to their works, not according to whether or not they believe in Jesus. When Jesus said all who don't believe in him are condemned already, he wasn't referring to hell itself, he was referring to the throne judgement.