Yes, but Lieberman would've sat in the Veep office for 4-8 years. Bloomberg (or any McCain VP) stands a good chance of moving to the Oval office rather quickly.
Bloomberg's politics will get him into more trouble than his religion. Similar to DC, you can't live in NYC without losing touch with the way off life of 90% of the rest of us in flyoverUSA. Much to the surprise of many a New Yorker, the rest of the nation is quite different in terms of values, desire to be more than 3 meters away from your nearest neighbor, own a 2k sqft house that costs less than $5,000,000, etc.
As a nation, I think we admired Giuliani's managerial skills, not his values or politics.
Shining in a crisis will get you a higher appearance fee, but might not extend your campaign past Florida.