It is quite possible to finish the game without getting all the Djinni. When I went though the game, I thought I was being quite thorough about getting into all the nooks and crannies to find all of them. Imagine my surprise, by the end of the game, I was missing about a half-dozen or so. It didn't impact my ability at all to take out the end boss.
Obviously, you don't get all the stat increases or classes, but that's more for completeness's sake than any requirement for the end boss.
Plus, it would be pretty bad game design to require every Djinni to beat the game, and then go and ensure that some of them are missable, thereby making sure you can't finish the game.