You have the first steps right.
You need to keep that oval shape you draw over your logo, select that circle and go in the menu "Object" and select "Expand".
That's probably what was missing. The pathfinder commands won't work with your thick stroke unless you convert it to a shape.
Then do your Pathfinder: Go on the Pathfinder panel, then select all, and you can use Divide (you can use trim too). Then you need to clean up what you don't need.
Then for the 2 round shape at the ends of your new shape, you can simply trace a perfect circle, and duplicate it at the other end as well.
And then select all and do another Pathfinder + Divide... and delete the white parts you don't need.
Is that the result you were hoping for?
Make sure your circles from the previous step are well aligned or you might need to fix the end curves manually.
If you want to create a crescent that ends up in a very small point, you can simply draw your first oval using NO stroke and a black filling.
And then trace a bigger and wider oval in white, and put iton top of that black oval.
Adjust it as you prefer! Sometimes it looks nice if the white oval is a bit more on one side, it gives some kind of dimension to the crescent.
And then do the steps above with the Pathfinder and Divide (and then clean up.)
You can also use "Trim" instead of Divide and delete the bottom part of what was "cut".
To delete the useless white and black oval parts, a quick trick is to use the direct selection tool (white arrow), and trace a rectangle that will include the middle paths of that oval... then delete!
You will still need to "clean up" your logo and remove the parts of the soccer ball that were cut.
When you "select all" your logo after all this, you should see something like this (your gradients will not look like this though):
If you want a gradient, you should do it in 2 separate steps... not directly as I explained above.
1A) Instead of using the oval with the stroke on the very first part of my answer, use a white filling oval. Duplicate it, you'll need it for step 2.
1B) Do your pathfinder + Divide on the logo.
2A) With the second oval, now add your thick stroke, do the "expand" command and then select that shape, and click "merge" on the pathfinder.
2B) Then you can "crop" again that oval in half, using a white rectangle or the 2 circles I mentioned above. And do another pathfinder + Divide on all this...
Then you can add a vertical gradient into this.