We are not saved by works, but all through the book of John, the admonishment is to believe on the Lord Jesus Christ for receiving eternal life. Not to mention other scripture also, where belief becomes before receiving eternal life. So therefore, the believing is led by Him, and not us.. and isn't a commitment to Jesus of good works or good behaviour, but a response to being convicted in the soul. It is not a contract we are giving our whole lives to Jesus.. and in return we are given eternal life. It is receiving His payment for our sins. A free gift. Not making Jesus 'Lord of your life'. But knowing you are a sinner and asking for Jesus to forgive your sin and give you eternal life, under conviction.
John 3:16, 3:36, 5:24, 6:40, 10:28. Romans 5:8, 10:9-10 etc..
Making Jesus Lord of your life and committing your life to Christ are good things, obviously nothing wrong with them, but they can only happen AFTER already being eternally saved, because before salvation.. there is nothing from US that is salvic. It isn't our commitment or our makings that get us saved.. but His forgiveness and then indwelling presence.
Jesus only intervenes with those whom He had chosen to salvation, not everyone. Should someone be "led by him", it is only because they already have been saved; the same is true with being "convicted". Therefore, neither brings them salvation because to be so, they were already saved.
Anything that someone must "do" to be saved, regardless of what the "do" may consist of, makes it a work. I agree that it is a free gift as you say it is, but for it to actually be one, there can be no preconditions nor requirements whatsoever associated to the receiving of it, otherwise, it wouldn't be free. This can be seen in the below verses. Observe in them that the "you" had been dead spiritually, and yet without any intervention nor action on their part, they were instantaneously translated from spiritually dead to spiritually alive (saved) by God. That instantaneous spiritual translation with nothing between the two spiritual states except God, makes it God's free gift.
Regarding the receiving belief before eternal life, I do not think that to be possible, because while dead in sin, the unsaved have no spiritual comprehension nor awareness to be able to believe, any more than a physically dead person is able to comprehend physical concepts, no matter how many times, nor how well, they might be explained to them - when God uses the word "dead" as He did in the following, He means truly dead, not figuratively dead. We can know they weren't meant as figurative, because we can see from that death, they were made alive.
[Eph 2:1 KJV] 1 And
you [hath he quickened], who were
dead in trespasses and sins;
[Col 2:13 KJV] 13 And
you, being dead in your sins and the uncircumcision of your flesh,
hath he quickened together with him, having forgiven you all trespasses;
This can also be observed in the following verses. In them we are informed that it is impossible for the natural (unsaved, spiritually dead, man) man to know the spiritual, and therefore, to believe until and unless they become saved. Therefore, believing before salvation cannot occur.
[1Co 2:13-14 KJV]
13 Which things also we speak,
not in the words which man's wisdom teacheth, but which the Holy Ghost teacheth; comparing spiritual things with spiritual.
14
But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know [them], because they are spiritually discerned.
It is confirmed by the following verse that to believe is given to someone on the behalf of Christ - it is not of/from themselves.
[Phl 1:29 KJV]
29
For unto you it is given in the behalf of Christ, not only to believe on him, but also to suffer for his sake;