> The Manhattan project was a government project that was run like a startup.

Run directly by the government employees, and directly hiring the nessesary people to work for the government. today we would call that socialist or some such, opposite of startup?

> If such a project happened today, academic scientists would be trying to figure out ways to bend their existing research to match the grants

You missed important step- such a project today would start by the government handing out grants, stimulus and tax breaks. If they directly hired the nessesary people, there would be no fighting for grants. People do what incentives demand