I think John Podhoretz captures it:
Given this record, Thompson has effectively proved what skeptics have been saying all along. He didn't want to be president. He doesn't like running for office. He doesn't have either a killer instinct or a ravenous hunger. And he really doesn't have a sense of mission. With all this in evidence, no Republican presidential nominee in his right mind would choose Thompson for his running mate. This isn't his game or his field or his love. We won't see Thompson with his arm raised at the nominee's side at the Minneapolis convention.
In 2006 I had a number of opportunities to see Thompson speak. He was an eloquent advocate of the war in Iraq, and I remember thinking at the time that if the president could only make the case the way Fred does, it would fundamentally change the way the war was discussed in this country. I also heard him talk about the problems within the intelligence community, which he understood thoroughly. I don't dispute Pod's analysis--it's hard to see what Fred would bring to the table in a campaign against Obama or Hillary. But if he's a liability on the stump, I tend to think he'd be a real asset in the Oval Office. Others seem to think that Fred's problem was mostly a matter of poor organization. While that may reflect poorly on Fred's ability to lead an organization, if you dropped him into a high-functioning campaign, perhaps he could play the part as well as he did in Hunt for Red October.