It was not just foreign countries that needed to be convinced by selecting more moderate leaders: there was also the Upper South, the Border States, Northern Democrats and conservatives and those in the CSA who weren't in favor of secession. I don't know how radical or moderate the convention or Congress that selected Davis was. It does tend to happen in revolutions that the most radical agitators who make the revolution are passed over when heads of government are chosen. It may be that cooler heads prevailed at Montgomery. Then again, it may be that pragmatic grounds motivated even some of the more passionate and enthusiastic secessionists to choose someone who was perceived as being more moderate.
It looks like Stephens, an old Whig, did oppose secession, but here we have Jefferson Davis's November 10, 1860 letter to Robert Toombs, written a few days after the election. Davis urges on purely practical grounds that the cotton states coordinate their actions, so that no state secede too early before it was clear that others would support it. Some arguments for caution and deliberation in late 1860 may just have been delaying tactics until Southern political leaders got all their ducks in a row, not indications of indecision, deep reflection or emotional torment. Doubtless, Davis felt regrets on leaving the Union, but he was also able to think quite cooly about what he and his state were doing.