"I was wondering how they were going to remove this one tile without damage to it and the surrounding ones but to my surprise she scraped out all the surrounding grout and started forcing what looked like a very liquid type of cement into the gaps and continued tapping until hollow sound disappeared. Is this a normal or acceptable and successful way to fix this type of problem. "
Had the same problem and they used the same way, but it didn't worked.You have to remove the loose Tiles, some concrete and fix it with Cement.