Fun games can certainly be made with this approach. It happens a lot. This is the model preferred by frugal publishers. It's cheaper, less risky, and can still end with a decent profit.
Great games though are never made this way. They start with the design, the genre will inevitably come after the game is finished.
-Shadow of the Colossus (Action/Adventure)
-Rez (Shooter-on-rails)
-Meteos (Puzzle)
-Gran Turismo (Racing)
-Metroid Prime (???)

Rez is hardly just a shooter-on-rails. People still squabble over exactly what Metroid Prime is. At the end of the day though it doesn't matter because they completely nailed a fun and interesting design.
Here's the question:
-What are some more examples of great games that disregard genre?