What kind of circle jig? Is it the one that pivots the entire router around a fixed center point, like for cutting round table tops?

The only time you need to have the router bit perfectly centered in the base plate is when using a guide bushing to trace a jig, like cutting dovetails. I center it visually during setup. Put the guide bushing on the base plate and the base plate loosely on the router base. Insert the router bit through the guide bushing and into the router collet and tighten. With the bit tight in the router, you can visually center the base plate by looking at the tiny gap between the bushing and bit.

Yes, the centering bit will work quite well.

The centering is required when changing base plates, since the manufacturer cannot guarantee the different base plate makers---you can even make your own baseplates----will get the plates centered.