Your claim that it is a false doctrine is inappropriate. It is not a false doctrine akin to heretics and false teachers, but is a topic worthy of discussion and even agreement to disagree.
More to the point of your post, you quoted the experience of Peter in an attempt to show that faith equals faithful exercising of that faith.
Just because one has the faith of God, does not equate that they are faithful to exercise that faith. Consider Elijah after the mountain top experience.
Peter had the faith of God, but had little experience and no practical knowledge as the Holy Spirit exercising that faith through a believer's everyday living. The Work of the Holy Spirit and the Word in the believer exercises and demonstrates to the believer the faithfulness of God, but also the reliance upon the new nature (including faith) that He has given.
As 1 Corinthians 4 states:
"4 I thank my God always on your behalf, for the grace of God which is given you by Jesus Christ;
5 That in every thing ye are enriched by him, in all utterance, and in all knowledge;
6 Even as the testimony of Christ was confirmed in you:
7 So that ye come behind in no gift; waiting for the coming of our Lord Jesus Christ:
8 Who shall also confirm you unto the end, that ye may be blameless in the day of our Lord Jesus Christ.
9 God is faithful, by whom ye were called unto the fellowship of his Son Jesus Christ our Lord."
And again in Galatians 2 states:
16 Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justified.
Ga 2:20 I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me.
The verses indicate that the believer does not fluff up their own faith, but lives by (the conduit) of the faith of Jesus Christ.
It is also important to know that the word "faith" is used throughout the NT as the persuasive credence in which one ascribes and adheres. There is no moral nor natural man ability to have such "faith" toward God, nor is there any unregenerate power in which one can express such. The Scriptures clearly state that is not the case of the natural man's state of affairs.
One further selection confirms the faith is not of human generational ability. Again in Galatians 3 it states:
"25 But after that faith is come, we are no longer under a schoolmaster.
26 For ye are all the children of God by faith in Christ Jesus."
That little underlined phrase (that faith is come) is more validity that it is not an innate in humankind, but given by God to the believer.
Also, the little word "in" is "en" which denotes a fixed position. NOT a preposition signifying movement in which one is placed into, but preposition of a decree, or stamp such as a king's stamp on a document making it official.
Does unregenerate man have a form of "faith?" 2 Thessalonians 3:2 says, no. If there is some human generated or innate faith it is fallen and has no desire to understand or know God, and God does not ever raise the fallen in glorification, for He has given us "new natures - created in Christ Jesus."
Your point that it gives man some "excuse" is merely a desire to validate your view with unsupported hyperbole and is not Scriptural.
But you already knew all this.