Do flight engineers also have to earn their pilot's license in case the pilot can't fly the plane?

Flight engineers are a dying breed in most airlines all over the world these days. Most air carriers all over the world have upgraded aircraft to where the engineer position is no longer needed as both pilots can handle duties formerly delegated to the engineer.
When I was hired, I was a fully qualified pilot but was given training by my carrier to become a turbojet flight engineer.
The airlines today always hire on the basis of a pilot being able to move up eventually to the Captain position. If memory serves, Pan Am and Eastern (now history) and perhaps others hired flight engineers as a permanent position with little chance to move up to a pilot seat. Many of those people were hired on their maintenance background and weren't required to hold a pilot certificate. There are likely still a few aircraft in the business world that still employ an engineer but very few. I would pursue my pilot ratings and add the engineer certificate only if you want to bolster your marketability.(a very expensive rating with few schools offering that rating training) It is not a rating airlines are looking for these days.