He would have played well in the NASCAR South and the union North - he had legitimate rail-splitting railroad-working cred. Yes, it’s part of the Lincoln Mythos, but the man was no stranger to physical labor.
I blame TV:
Nowadays, there is a limited amount of news time on TV. The news organisations devote most of their time to the bigger national and international stories, and only a little for the local stories.
Back in 1860, this situation was the opposite. More local news, less “big” news. Newspapers were also more locally owned, I imagine. Large newspaper companies were in existance, but I bet that compared to the independent newspapers, the circulation numbers were still smaller.
Smaller papers back then (in Illinois) would be reporting what State Rep Lincoln was doing for his community, but today, we get Paris Hilton. Heh.
Off the top of my head, I could not tell you who the State Rep is for my district. But Sen Boxer, Feinstein, and Rep Pelosi are much more reported upon.
(Am I making any sense?)
Lincoln would be too old (199) and out of touch. He almost certainly had never used a computer.
The Constitution has a minimum age, not a maximum age, for Presidents. And do you think the incumbent can use a computer?
As I recall, he didn’t do too well there the first couple times he ran. 
That’s because he didn’t put a “3” sticker on his carriage. He’d know better now.
Edwards had the magical title of Senator, though, which has always conferred greater credibility for a presidential run. Lincoln’s only term was in the House, not the Senate, and it had ended 12 years previously.
I think a politician with that thin a resume today would encounter greater resistance on the experience issue.