Actually Total depravity is a Biblical doctrine. Lets first turn to Romans 3:10-18 and The passage clearly teaches that no one seeks after God for they all seek their own ways and not of God.
Next turn to Matt 16:17 and Jesus teaches that it was God that revealed to Peter the fact that Jesus is the Christ. Next turn to John 10 and Jesus speaks about his elect and they are the ones that hear his voice. Jesus did not say that the non elect hear him, He clearly spoke of His sheep whom are the elect. Next turn to John 6:65 and the text says that no on can come to the father unless the father has enabled him (NIV). Likewise Mt 11:27 says that the son chooses to reveal himself to some and that some is the elect. For the icing on the cake turn to Jonah 2:9 and the text reads "Salvation comes from the Lord" (NIV). Therefore it seems obvious that the Holy Spirit chooses to reveal himself to some and those that are of the elect. Regarding the passages that you mentioned Jesus was preaching for the elect to respond to salvation for He knew that not everyone would do so. Likewise I try to follow His example when I preach for I know that only the elect will respond to the call to salvation whom God has prepared.
I can understand your frustrations about this doctrine for I once thought as you did. I know its a hard one to understand.
It's not difficult to understand at all, it is just wrong.
If I were to say that none of my neighbors ever goes to church, no, not one, would you understand that to mean they are unable to go to church? NO, and no reasonable person would ever assume that, yet that is exactly what you are doing with the scripture in Romans 3. The fact that no one seeks God does not mean they are unable to seek God, you are reading Total Inability into the text when it is not there.
Mat 16:17 certainly does not teach inability, how do you get the doctrine of inability out of this verse? Jesus simply says that the Father had revealed to Peter that Jesus was the Christ. Peter was ABLE to learn from the Father, so if anything, this verse teaches ability, not inability.
Jhn 6:65 does not say no man can come unless God "enables" him as you say.
Jhn 6:65 And he said, Therefore said I unto you, that no man can come unto me, except it were
given unto him of my Father.
Jesus said no man can come to Jesus unless it were "given" unto him of my Father. What is he talking about? What was given by the Father? Well, Jesus had just spoken of his words, this is what is given.
Jhn 6:63 It is the spirit that quickeneth; the flesh profiteth nothing:
the words that I speak unto you, they are spirit, and they are life.
64 But there are some of you that believe not. For Jesus knew from the beginning who they were that believed not, and who should betray him.
It is the word of God that is given to men, but men must believe, listen and learn from God's word before they will ever come to Jesus. This is what Jesus had taught in verse 45;
Jhn 6:45 It is written in the prophets, And they shall be all taught of God.
Every man therefore that hath heard, and hath learned of the Father, cometh unto me.
Every man that has heard and learned from the Father comes to Jesus. Those who refuse to hear and learn will not come to Jesus.
Mat 11:27 is not teaching that Jesus reveals the Father to only some men, read the following verses:
Mat 11:27 All things are delivered unto me of my Father: and no man knoweth the Son, but the Father; neither knoweth any man the Father, save the Son, and he to whomsoever the Son will reveal him.
28 Come unto me,
all ye that labour and are heavy laden, and I will give you rest.
29 Take my yoke upon you,
and learn of me; for I am meek and lowly in heart: and ye shall find rest unto your souls.
Yes, in verse 27 Jesus says no man knows the Father except those he has revealed the Father to, but who does this apply to? Read the very next verse, Jesus says, Come unto me,
ALL YE that labour and are heavy laden. So the offer is to
ANYONE. He offers to give everyone rest, and in verse 29 he tells these persons to "learn" from him, and they shall find rest. Jesus is certainly not limiting his offer to only a few persons here, but any man who will come to him. It certainly is not teaching inability.
And Jonah 2:9 certainly does not teach inability, it just states that Salvation is of the LORD. How does that teach inability?
So, your view is not difficult to understand, it is just wrong. None of these scriptures is teaching inability or Limited Atonement, you are reading that into scripture.
When you preach the gospel, it applies to all men. You should be glad to know that. You don't have to hope some "elect" person comes along, any man that hears you can believe if he chooses to do so. And if he listens and hears, he can come to Jesus, it was given to him by the Father through you.