Titus 2:11
11 For the grace of God has appeared, bringing salvation to all men, 12 instructing us to deny ungodliness and worldly desires and to live sensibly, righteously and godly in the present age, 13 looking for the blessed hope and the appearing of the glory of our great God and Savior, Christ Jesus,
I don't see in this passage that you referenced where it says Man is the source of the belief (and only some Men excercise the belief). I see the same "salvation to all men" reference that we alluded to in 1Timothy 2:5-6. Again, this scripture implies a universal salvation.
I think the topic might better be addressed if we grapple with the issue of John 3:3. What does it mean to be "born again"? Is this something that results from Man's (self inspired) belief? Or is this something that occurs which allows Man to believe?
I didn't say that. I said God has provided the grace that all men might believe. Without that grace, Man cannot come to Christ on his own.