It's basically too late, anyway. He would never be able to build the momentum needed to win any primary (Dem or Rep). Plus, he's not really a likeable person, and some level of charisma is necessary to win any political race.
The idea wasn't that he'd run in a primary, but that he would mount a third party 'no labels' bid in the general election. Polling made it really clear that this would throw the general election to Trump as some Biden voters would shift to Manchin, but few Trump voters would.
Run for president on what platform? Genuinely, was he planning on running against Trump or against Biden?
Because both parties see him as a traitor. To Democrats, he was the lone senator blocking pivotal congressional action for years. To Republicans, he's been sleeping with the enemy and pushed out what would have otherwise been a guaranteed seat in the senate so he could be a "compromise Democrat".
He was being courted by the No Labels group. He wasn't going to win against Trump or Biden, but might have siphoned off several percent of voters who would otherwise go for Biden, throwing the election to Trump. Which was the whole point of trying to get him to run.
Thank you for the actual answer. I still think that's really stupid though, as you'd have to be a fool to vote for him on his record in congress. I think it's just as likely he'd siphon votes from Trump in that case.