Okay,
First, I like the Zephram Cochrane in FC, seems actually perfect in sync with TOS. I mean, just look at the embarrassment of TOS guy when they talk to him about his amazing deed. Tie it in with his horror back in 2063, what he was like back then, and imagine what TOS Cochrane must have been thinking, *Yeah, yeah, yeah, god if you knew what I was like back then, you'd cock me a good one instead of all this praise.*
Also, this whole think about Cochrane being the great engineer is rather not true. He had an entire team of engineers with him to build that ship, if you remember, he was just the theoretical guy who figured out how to do the warping thing, and was probably involved in the practical design with his lead designer, but he wasn't mister super engineer.
As for why it was like that; perfectly explainable. The nuclear holocaust mentioned in Bread & Circusses, probably the war in which Colonel Green did his ethnic cleaning; which apparently by historians was renamed WWIII after TOS, and the Eugenics Wars, previously WWIII now just the Eugenics Wars (which wasn't all that long ago from that point either), had just occurred. Earth is in shambles. I doubt anyone would be exactly happy at the time, and being a drunk isn't that far fetched; hell, if there is a good reason to be a drunk, I suppose it has to be living in a nuclear waste land.
It would also rather explain Cochrane himself working on the engine, probably with funds from some rich people wanting to get off that nuclear waste land, and since Cochrane is the only one who understands the physics behind the warp drive, he'd have to do the hiring and leading of the engineering team that puts the theory into practice.
It fits rather well to me, and I love FC as a movie.