A better approach and the one used by most chimney repair masons who are not going to simply reline the flue, is to remove all of the damaged materials down to sound chimney flue liner and brick or block, then rebuild with new clay flue liner and brick (or block), adding an appropriate...
The repair job you describe sounds very similar, in process and cost, to the one described here CHIMNEY RELINING REPAIRS You'll see the process and tools described are similar to what you've been told. Might be a little easier for the chimney repair crew to complete the project in ...