You probably don’t want to use strokes at all. Just shapes. Strokes are basically an effect. A logo wants to be made out of a single shape object.
A key concept in vector graphics is combining shapes. For example, subtracting smaller shapes from a larger shape to make holes in the larger shape. If you think of the logo you are trying to recreate as having originally been made out of a single circle of paper, you can start by drawing that circle and then creating smaller shapes to subtract from the circle to make the holes you want.
For example, here is a black circle with a gray circle sitting in front of it, with no stroke on either:
… we can Subtract the gray circle from the black circle, and that will leave a hole in the black circle the size of the gray circle:
… so that we now just have a single path object that is a thick circle. You can do a lot more with that than you can with a circle object that has no fill and a very thick stroke.
In the case of the logo you are trying to recreate, you would want to create a sort of a J shape and punch that through a larger shape to get the J-shaped holes that you want. You can make the J shape by combining smaller shapes, such as a rectangle and a circle. To get the cut in the bottom of the J, you can subtract a small circle from it.
So the thought process is one of manipulating paper shapes and cutting holes in them and gluing them together until your result is a single path object that has fills but doesn’t necessarily have strokes.
Also, since the logo you are making has symmetry, you might want to just create half of it, then duplicate and mirror that half and combine it with the opposite half to create your final logo. You can also create it at first so that the symmetry line is vertical, and then in your very final step, rotate it to give it the offset that you want.
This Adobe document might be helpful:
How to combine objects in Illustrator