I don't think I could argue that such things "have nothing to do with the repentance Christ gave us." The verse you quoted says that we will be saved if we confess and believe. Those seem important right?
If we stay strictly with Scripture, the New Testament repeatedly connects receiving the Holy Spirit with believing in Christ, not with first completing a process of Christian obedience.
Paul asks the Galatians:
"Received ye the Spirit by the works of the law, or by the hearing of faith?" (Galatians 3:2)
His expected answer is: by faith, not by works.
Likewise:
"That we might receive the promise of the Spirit through faith." (Galatians 3:14)
And:
"In whom ye also trusted, after that ye heard the word of truth, the gospel of your salvation: in whom also after that ye believed, ye were sealed with that holy Spirit of promise." (Ephesians 1:13)
Notice the sequence: hear the gospel → believe → be sealed with the Holy Spirit.
Paul also says:
"If any man have not the Spirit of Christ, he is none of his." (Romans 8:9)
That implies there is no category of true believer who belongs to Christ but has not yet received the Spirit.
The account of Cornelius is often cited because the Holy Spirit came upon the Gentiles while Peter was still preaching:
"While Peter yet spake these words, the Holy Ghost fell on all them which heard the word." (Acts 10:44)
Why was there no repentance?
"God, which knoweth the hearts, bare them witness, giving them the Holy Ghost, even as he did unto us... purifying their hearts by faith." (Acts 15:8-9)
Again, faith is emphasized.
As for whether someone must live the Christian life in order to finally become saved, passages on the gift of salvation are very strong:
"To him that worketh not, but believeth on him that justifieth the ungodly, his faith is counted for righteousness." (Romans 4:5)
"Therefore being justified by faith, we have peace with God through our Lord Jesus Christ." (Romans 5:1)
"Verily, verily, I say unto you, He that heareth my word, and believeth on him that sent me, hath everlasting life, and shall not come into condemnation; but is passed from death unto life." (John 5:24)
The tense is significant: "hath everlasting life" and "is passed from death unto life."
At the same time, the New Testament also teaches that genuine faith produces fruit:
"For we are his workmanship, created in Christ Jesus unto good works..." (Ephesians 2:10)
But notice the order. Ephesians says:
"For by grace are ye saved through faith... not of works" (Ephesians 2:8-9)
and then
"created in Christ Jesus unto good works" (Ephesians 2:10)
The works follow salvation; they do not purchase it.
Does a person receive the Holy Spirit when they believe, or only after they have sufficiently lived the Christian life?"
The weight of passages such as Ephesians 1:13, Galatians 3:2, Romans 8:9, and Acts 10:44-48 points to receiving
the Spirit upon believing. The Christian life is presented as the result and evidence of God's saving work, not the
prerequisite that earns it.
Did you repent of your sin before you called on the name of Jesus for salvation.
If so, you cannot repent without the Holy Spirit illuminating you to your own fallen state.
Therefore, how can someone repent before receiving the Holy Spirit?
If repentance was practiced by you after receiving the Holy Spirit then that is during your Christian life,
not at the time of salvation.