Making your own engine is definitely a good learning experience, but it’s not required for making a game. Game engines are tools for making games, they don’t make the game for you.
You don't even have to "make your own game engine" - often you just take a windowing library, optionally with some barebones renderer (like SDL or Allegro) and simply make a game without an engine. This approach was very common at the beginnings of compos/jams like Ludum Dare, it only changed somewhat recently with increasing popularity of ready-made engines.