A bit of an exxageration, but you could take Lattimore at 1 overall this year, and even though you would get some weird looks, no one could really argue against it. Obviously the 2 injuries are an issue, but aside from those, you have no other issues with him. At 21 years old with his work ethic, I'll put my money on a full recovery for him.
I know he is going in the 1.10-2.02 range, but tell me who is going before him that doesn't have a big question mark of his own. You want your first round pick to be as sure a thing as possible. Thanks to all the coverage Lattimore has received, we know he has the character, the motivation, the work ethic and the skills to be a sure thing. All the other guys have question marks regarding their skill set, character, size, how their team will use them etc.
I know the guys going before Lattimore seem like a safer option because of his injury, but I think he may still be the safest option in this draft. To answer the OP's question, I absolutely think a team who is rebuilding should take Lattimore at 2.01.