I don't think any other then Santoram really are. I think Gingrich's discovery of religion was conveniently timed after his last sex scandal. I think Romney uses Mormonism as a great fundraising tool, but his views on abortion, or stem cells sound like TBM answers, and he has remained largely silent on the subject of gay rights during this campaign.
As for Huntsman, I didn't even know he was a person until a couple weeks ago, but I know a lot of guys here don't think he is very TBM.
"It is a truism that almost any sect, cult, or religion will legislate its creed into law if it acquires the political power to do so." (Robert A. Heinlein)
I just wish American politicans were less in-your-face about religion (although the same can be said for Americans in general). And where are the moderate Christian politicians? Where are the moderate Christian Americans? Why are the vast majority of American right-wingers (politicians and average people) so in your face and extreme? I'm not saying a prominent politician should be an atheist, but they should be secular and even if they are Christian they should be secular and less in your face about their faith.
The irony is that to the social right, their greatest hero was Ronald Reagen, a man who could not be bothered to go to church during his administration, and who could barely be called religious at all. The religious right is something of a new thing. One that really annoys me. My own position is that I am a libertarian on the social and economic issues, but I think that party is weak in foreign policy.
I think it does to a degree...and it could make the difference. I had a friend in college who was southern Baptist from a rural town not too far from Atlanta. He became visibly upset after he saw my parents after they came to visit one weekend. He tried to be nice about it but kept going on about how the races shouldn't mix and so on. Mention the Pope, Mormons, or Islam and he reacted the same way. Since most of the Republican party now is made up of white evangelical Christians it will dampen the enthusiasm somewhat. In their view Catholicism is bad but Mormonism is worse and they will reluctantly go to the voting booth -- but vote while holding their noses.
One thing I could never figure out about Mormonism was the complete lack of empathy (from the religion -- not individual people). You would expect the Brethren to say things such as "our Prophet was lynched, our people were persecuted so therefore we have something in common with you" and so on but they don't do that. I could be wrong but I read speeches and watched video from conference and fireside talks and I could not find any statement from them to that effect.
I wouldn't be too troubled if Newtie or Santorum were kennedy-style Catholics.
But Santorum is one of the Catholic League Wackadoodle fanatics, and Newt is a grandstander extraordinaire who will do and/or say anything for votes and attention.
Kinda like Mitt.
Huntsman is an unknown quantity, and for that reason alone he scares me even more than the other 3.
They have to spread the appeal to capture more votes. So Romney would never choose Huntsman because the common Mormonism is too obvious of a lightning rod (not to mention their political differences).
Even Gingrich-Santorum probably wouldn't be considered.
If Romney wins he'd be wise not to pair up with any of them, but find a likeable genuine conservative to counter his right wing apologetics.
On the OP issue of whether his (or any of the others) faith matters, I would say that what obviously matters more than their personal faith, is the public perception of their faith. It is a sad and unfortunate fact (and I speak as a right winger) that to get nominated in the GOP, you need a Christian faith and anti-scientific views on the origins of man and the universe.
It's a sorry state of affairs when the candiate with the most sensible views on gays and evolution is a Mormon.