In general, people totally miss what that phrase is about. Roughly it means “I am a person of faith. I have thought about this, I have prayed about this and I have come to the conclusion that I’m going to run. The inference is that if God didn’t want it, said prayer would have led him to a different conclusion.
It does NOT mean that God/Jesus appeared to him/spoke directly to him and said “Run”.
Now as for public perception, people either accept people of faith as leaders/Presidents or they do not. The language POF use is part and parcel of that faith. It’s up to the public to accept or reject it. Because if a person actually believes it, he/she isn’t going to say whatever makes people happy for their votes.
And the politicians that Cain/whoever would have to work with can deal with it. They don’t have a choice once the people say “We want him” or whatever person of faith.
Politicians do this all the time in order to appease. Obama said he was inspired by (some) god(s) in a lot of Christian Churches when he was campaigning. The guy even claims to be a Christian who, by default, is led by Christ in the first place.
When politicians invoke God, the statement of faith better be shown by their actions or they are just very weak and often are frauds.