He didn’t say that in the context that no one can choose to believe in Him. Nothing in the entire chapter even remotely indicates what your suggesting. In John 6 many of His disciples left Him and didn’t follow Him anymore. Why did they believe then turn to unbelief? Were they chosen by God then rejected? Apostasy is mentioned several times throughout the New Testament.
“At that time many will fall away and will betray one another and hate one another. Many false prophets will arise and will mislead many. Because lawlessness is increased, most people's love will grow cold. But the one who endures to the end, he will be saved.”
Matthew 24:10-13 NASB
“Therefore, do not throw away your confidence, which has a great reward. For you have need of endurance, so that when you have done the will of God, you may receive what was promised. FOR YET IN A VERY LITTLE WHILE, HE WHO IS COMING WILL COME, AND WILL NOT DELAY. BUT MY RIGHTEOUS ONE SHALL LIVE BY FAITH; AND IF HE SHRINKS BACK, MY SOUL HAS NO PLEASURE IN HIM. But we are not of those who shrink back to destruction, but of those who have faith to the preserving of the soul.”
Hebrews 10:35-39 NASB
“Those on the rocky soil are those who, when they hear, receive the word with joy; and these have no firm root; they believe for a while, and in time of temptation fall away.”
Luke 8:13 NASB
“At that time many will fall away and will betray one another and hate one another. Many false prophets will arise and will mislead many. Because lawlessness is increased, most people's love will grow cold. But the one who endures to the end, he will be saved.”
Matthew 24:10-13 NASB
“But the Spirit explicitly says that in later times some will fall away from the faith, paying attention to deceitful spirits and doctrines of demons,”
1 Timothy 4:1 NASB
“Take care, brethren, that there not be in any one of you an evil, unbelieving heart that falls away from the living God.”
Hebrews 3:12 NASB
“I am amazed that you are so quickly deserting Him who called you by the grace of Christ, for a different gospel;”
Galatians 1:6 NASB
There is no scripture that states a person can choose to believe in God, yet the Word of God is not silent on the matter of (1) the source of belief and (2) the One that does the choosing and (3) the ones that do not do the choosing.
In John 6 the learners (disciples) that left are not identified specifically as believers, in fact, they are identified as unbelievers because they did not believe in Who Jesus said He is by His words.
In Matthew 24:10-13 Lord Jesus used the words "fall away" which is not the same as "fall out of"; in other words, the ones that "fall away" on not on inside.
Hebrews 10:35-39 indicates nothing about choosing to believe towards God.
1 Timothy 4:1 contains "fall away" like Matthew 24:10-13, not "fall out of".
Galatians 1:6 with respect to Paul using the word "called", Lord Jesus says "Many are called, but few are chosen" (Matthew 22:14).
Hebrews 3:12 actually clarifies the "fall away" phrase contrasted to "fall out of" phrase with this phrase in Hebrews 3:12 "an evil, unbelieving heart that falls away".
I have wondered about Luke 8:13, so I thank my Lord and Savior Jesus for this correspondence. Let us examine the Word of God.
EXAMINING LUKE 8:13
"Those on the rocky [soil are] those who, when they hear, receive the word with joy; and these have no [firm] root; they believe for a while, and in time of temptation fall away"
The first clause "Those on the rocky [soil are] those who, when they hear, receive the word with joy".
This is clear, they received the word with joy.
The second clause is dependent on the first, but the NASB translators added the word "[firm]", so it should be removed for clarity, and here is the second clause "and these have no root". Jesus is the Root of Jesse (Isaiah 11:10). Jesus is the Vine (John 15:1).
The third clause is also dependent "they believe for a while", but Jesus did not specifically state that they believe in Jesus for a while.
The fourth clause is also dependent "and in time of temptation fall away" which contains "fall away" not "fall out of".
In all of this, Lord Jesus remains consistent with His words which include "My sheep hear My voice, and I know them, and they follow Me; and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand. My Father, who has given [them] to Me, is greater than all; and no one is able to snatch [them] out of the Father's hand" (John 10:27-29).
Not a single verse you cited indicates that a person chose to believe towards God.
Now, behold the salvation of the Lord Jesus, the Power of God:
God produces divine choice of we believers unto salvation, for the Christ of us Christians says
"you did not choose Me, but I chose you" (John 15:16) and "I chose you out of the world" (John 15:19)
SO, clearly, Jesus' words in John 15:16 and John 15:19 state God exclusively chooses us believers by/of/through God
God produces believer's faith/belief in Lord Jesus, for the Christ of us Christians says (see also a word about belief/faith (Greek ?????? pistis) and believe (Greek ??????? pisteuó))
"This is the work of God, that you believe in Him whom He has sent" (John 6:29)
SO, clearly, Jesus' words in John 6:29 state for us believers to believe in Jesus whom the Father has sent is by/of/through God
God produces believer's fruit of the Spirit/righteous actions/good works, for the Christ of us Christians says
"he who practices the truth comes to the Light, so that his deeds may be manifested as having been wrought in God" (John 3:21)
SO, clearly, Jesus' words in John 3:21 state fruit in we believers is by/of/through God
God produces believer's birth by the Holy Spirit, for the Christ of us Christians says
"Truly, truly, I say to you, unless one is born of water and the Spirit he cannot enter into the kingdom of God. That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not be amazed that I said to you, 'You must be born again.' The wind blows where it wishes and you hear the sound of it, but do not know where it comes from and where it is going; so is everyone who is born of the Spirit" (John 3:5-8)
SO, clearly, Jesus' words in John 3:5-8 state we believers being born again is by/of/through God
There is no level that a person can choose Lord Jesus because He said "you did not choose Me, but I chose you" (John 15:16) - Jesus, being God, did not provide any exception for choosing toward Jesus. Lord Jesus speaks to all believers in all time because He also said "I do not ask on behalf of these alone, but for those also who believe in Me through their word" (John 17:20)! All these words of Jesus are at the same supper! All glory is God's! With man, salvation is impossible (Matthew 19:25-26)! All glory in the salvation of man is God's (John 15:5, Isaiah 42:8)!