I see what your saying. I don't think at this conference Bill was talking to an audience of people who are struggling to believe that there is a God. This was a conference of believers, not a church service.
So the word "believe" becomes more about having faith for an answer to prayer.
I went back and listened to the segment again and here is a bit more of his talk. He is talking about his experience and the experiences of his members.
Faith does not come through striving. I have seen so many sweating and striving, saying, "I am fighting to believe". Faith comes not from striving but from rest. It is not the result of effort but the result of surrender. Its the most normal thing for a believer to believe. Its when we feel provoked by a lie that we have to strive and fight for something. Then we can really loose traction and loose so much of what God has established for us. And then we can beg God for a breakthrough that he has already been promised.
It is out of rest that we learn to recognize presence, and actually live from presence toward the situation. When you act out of a burning conviction and realization of the very nature of God then we can be like Jesus.
Striving never ends well. Its much better when I can turn my heart of affection for him towards him and bring my awareness to his presence that so quickly responds and rests on me and then live out of that presence towards the situation. I can't afford to let any problem, no matter what it is get bigger in my awareness then my consciousness of God's presence upon me. If I do then I will live in reaction to the problem instead of staying aware of his presence and living life from that awareness.
Your faith will always operate in the playing field of your revelation of his goodness. Its your understanding of his Goodness that invites us to exercise faith and pursue answers to prayer.