Do I need to go looking for the missing Fun Fun Island wheel?

The owner of Fun Fun Island is very sad. It appears he has lost his wheel. I'd like to help him out. I suspect it fell somewhere in the Lanayru Desert, the closest surface world. Do I need to go explicitly look for this wheel, or will it turn up in the normal course of my adventuring?

(A generic hint about its location would be nice, too, before mentioning the specific location. Please markup hints as spoilers.)

Finding it is not a requirement to finish the game. It just grants you access to the money making minigame that is covered in this question. I would suggest getting it as it is an easy source of rupees with little to lose and a lot to gain with relative ease.

Unless you are a very throurough in searching every spot there is in a map for anything, you probably wouldn't see this in your normal, go-for-the-objective adventuring. It's location is near a "hidden" chest and off the beaten path.

Hint:

To reach it involves some ledge climbing and the help of a Timeshift Stone.

How to get it:

When you first enter Lanyru Desert (after exiting the mine), there will be a Timeshift Stone nearby. Hitting it will reveal some vines on a nearby wall that you may climb up.

There will be a ledge that you can climb up to to reach another cliff on the other side of the little gorge.

You will end up on a narrow land bridge where below you will see an isolated chest sittng by and the wheel at the end (in the corner). You can't miss it.

It's a side quest, so it is entirely possible you may miss it, although eventually you will be able to dowse for it (you do have to complete a few other things first).