Correct. Saving faith is not a gift. It is generated by the hearing or reading of the Gospel, while the Holy Spirit convinces that person that the Gospel is true.
This question seems to me to be at the very core of the issue of all these "does God choose us or do we choose Him debates".
Maybe a bit repetitive but a little about me in these regards. I do not, nor never have, identified as a Calvinist. I had never read a single word John Calvin wrote before coming here about a dozen yrs ago. I am now very familiar with the 5 points of Calvinism, commonly referred to as T.U.L.I.P. I agree to some extent with most of them.
I was saved in a Calvary Chapel that could hardly be considered Calvinistic. For the past 12 yrs I've attended a Nazarene Church which is impossible to mistake as a Calvinistic Church. My Pastor and I disagree on a great many doctrines, but I firmly believe "in essentials unity, in non essentials liberty, in all things love". And my Pastors messages on Grace, and how to apply it to our lives are incredible, and something I need.
So no one "taught" me that God chooses us, and NONE of us would willingly choose Him first. Reading Scripture, and searching my heart, through the Holy Spirit got me there.
Now on to the faith question. First, I think the amount of twisting one has to do to make Ephesians 2 NOT say that faith is a gift from God is amazing.
So WHERE does faith come from?
You rightly refer to romans 10: 16-17
16But they have not all obeyed the gospel. For Esaias saith, Lord, who hath believed our report? 17So then faith [cometh] by hearing, and hearing by the word of God.
But verse 17 actually means our faith, put inside us by God, is ACTIVATED by the hearing of the Gospel. God "quickens" us, or brings us to life THROUGH that faith, God gave us, that is activated by the hearing.
Here is the Greek word for "cometh by":
1537. ek or ex
from out, out from among, from, suggesting from the interior outwards.
So we literally have faith, put there by God, that is dormant, and then drawn out by the hearing of the Gospel.
If we don't believe that then we must ask the following questions;
Are we better than others that hear the Word yet reject Him?
Are we smarter than those that hear the Word yet reject Him?
Are we more moral, and less of a sinner than they that hear and reject Him?
So no matter how deep we dig, God is ALWAYS there as the total and complete reason we believe and are saved.