Until recently your contractor would have been correct, but that is no longer the case. There is an economically pleasant alternative that will save you about half the cost of removal and replacement. In large commercial projects where concrete floors often are placed in sections, it has been standard practice for years to raise shifted or settled concrete sections, rework the earth below (usually the cause of the problem to begin with), and put the concrete back, creating a new level finish.
A fellow in Napa, California, named Alfred Allen, further developed this technique specifically for residential concrete flat work repairs (patios, sidewalks, etc.) Appropriately named the Allen Process, the method uses powerful air bladders to raise concrete without breaking it. The ground under can then be leveled and compacted, and the concrete is resituated. Once done, no one would ever suspect shifting had occurred. To the best of our knowledge Allen is the only fellow in the nation who does this. We have seen the results of his work, and it is amazing.