Given the propensity of legislators/executives/courts to construe the language of the Constitution and amendments as broadly or narrowly as desired, I doubt that removing the Bill of Rights would make any significant difference. See the Commerce Clause and its construction since the 1930s for an example of extremely broad construction, and the Necessary and Proper Clause of Article I for an extremely strict construction. Commerce can restrict what you grow in your garden for your own consumption. Necessary and Proper, despite being a literally broad clause, has been considered an insufficient ground for pretty much anything (unless combined with some other power).
For the Bill of Rights, the 9th amendment has generally been interpreted to convey no rights at all, while the 1st and 4th have been given a nebulous privacy component. Fun with courts!
TL;DR: The Constitution is very malleable to the whims of government, with or without the Bill of Rights.