By your definition of Total Depravity, is the gospel message sufficient as the preliminary step in reaching the lost; such that, a sinner can now accept or reject its message at this point?
If man could of his volition chose to repent and believe in God at any time during the conversion process, do you feel that this will cause a share in the glory of his salvation between God and Man?
If man shares in the glory of his salvation simply by believing, then we would be carelessly shifting the source of our justification onto faith itself rather than understanding it is 100% given by God alone. For a believer does not wield faith as the mechanism upon which he justify's himself, rather God has sovereignly promised and decreed to do 100% of the salvation work upon those whom have faith in Jesus' Christ as the condition for redemption.
It is by this condition of what would have otherwise been a powerless act of faith that makes the redemption so glorious and why faith is not meritorious as Paul so clearly lays out. For the power of faith is not in faith itself, but rather from God who uses faith as the mechanism to redeem humanity.