A, hopefully comprehensive answer to your question:
HCF listed MC cable is a code accepted method of accomplishing what you hope. However, there are often State authorities to don't allow it, depending on the application. For example here in Florida we have Florida Agency for Healthcare Administration (AHCA) that inspects all healthcare facilities. In inpatient hospitals, they don't allow MC. So, in this case, opening the wall is your only choice.
Also in many cases the specifications call for conduit only. In this case, it would be up to the GC (as long as you covered yourself regarding cutting patching and painting) to try an convince the Specifier, or customer to allow a variance for the use of HCF listed MC in this case, to open up the wall.
Lastly regarding fire rated walls. UL listed wall assemblies for 1 hour or more rating, require boxes to be physically secured to the framing. I.E. no cut in boxes. There are ways to accomplish this, but the easiest way is to open up the wall where the box goes. This is something to pay attention to in all types of construction, because it is not specific to healthcare.