No reason whatsoever.
Believe (believeth in the KJV) is the Greek work pisteuō and means:
1) to think to be true, to be persuaded of, to credit, place confidence in
a) of the thing believed
1) to credit, have confidence
b) in a moral or religious reference
1) used in the NT of the conviction and trust to which a man is impelled by a certain inner and higher prerogative and law of soul
2) to trust in Jesus or God as able to aid either in obtaining or in doing something: saving faith
3) mere acknowledgment of some fact or event: intellectual faith
2) to entrust a thing to one, i.e. his fidelity
a) to be intrusted with a thing
it is also from the root word pistis which means:
1) conviction of the truth of anything, belief; in the NT of a conviction or belief respecting man's relationship to God and divine things, generally with the included idea of trust and holy fervour born of faith and joined with it
a) relating to God
1) the conviction that God exists and is the creator and ruler of all things, the provider and bestower of eternal salvation through Christ
b) relating to Christ
1) a strong and welcome conviction or belief that Jesus is the Messiah, through whom we obtain eternal salvation in the kingdom of God
c) the religious beliefs of Christians
d) belief with the predominate idea of trust (or confidence) whether in God or in Christ, springing from faith in the same
2) fidelity, faithfulness
a) the character of one who can be relied on
So no, there is no reason to add/change any words here to promote a false works based salvation.