The Hineni prayer is said by the Chazan before the silent Amida on the high holidays. In it, the Chazan admits his personal flaws but asks still to be allowed to be a righteous and competent representative of the kahal.
Why then, is it recited before the silent Shmoneh Esrei when the Chazan is not yet acting as a representative of the people. The Chazarat Hashatz is the time when the Chazan is leading the communal prayer and becoming the spokesperson (as evidenced by both the justification for Chazarat Hashatz during the year and prayers such as misod chachamim, heyeh im pipiyot and ochilah lakel).
Shouldn't it be said right before the repetition?