Is the dentist supposed to ground off the porcelain on top of the metal crown in order to get it to fit better?

You’re experiencing a classic error with dentists where they did not shave your natural teeth down enough because they were trying to be conservative with taking your natural teeth structure away. By doing so, there wasn’t enough room for the laboratory to make crowns with minimal thickness requirements. In normal situations, a dentist should’ve ground your teeth down to create X amount of space so the lab technicians can make the proper thickness crowns. If there isn’t enough space for the lab techs to make their crowns then they’ll make the crowns with the normal thickness anyways and let the dentists to deal with the bite issues. To make the matter worse, if the dentist doesn’t communicate with their labs and the crowns will definitely come back with a standard required thickness. As result, when your dentist cements your crowns in, they are way too thick meaning you’ll bite and hit them first. Only fix at that time and start to grind on the “too high” crowns and there goes the porcelain layer. Dentists will often blame the lab techs for making the crowns too thick while in reality they don’t look in the mirror at themselves to evaluate or ask themselves “did I provide enough clearance for the lab?” If you don’t like seeing metal on your new crowns, who would? Then you’ll need to go back to your dentist and have those crowns cut off and have your dentist reduce your teeth more (create enough clearance) and order new crowns. But be warned, if your dentist don’t have the skill or knowledge to preserve enough tooth structure (ground too much away), then no crowns will stay on. So there is this perfect happy medium, not too much ground away but just enough taken away so the lab has enough space. Also, instead of getting porcelain-fused-to-metal (PFM) crowns, ask to get “all-ceramic” crowns like a BruxZir crown. PFM are old school with metal underneath and they’re less desired and used nowadays. 

A well prepared tooth and a properly made crown, should NOT REQUIRE so much adjustment as to expose 50% of the underlying metal. You should consult with that Dr. and express your concern.
You should also know that there are crowns that can be made today that do not have ANY metal, and are strong and reliable- and in most cases, look better than the metal based crowns.