Any short cuts?

Before I spend the time to use the pen tool in Illustrator to trace every gemstone . . .

At most, there are maybe only 6 different cuts there, so yes, tracing each and every stone would be silly. And, the thing is symmetrical (or, your perfected vector version of it should be), so no matter how you trace, you'll only need to produce half the layout, then duplicate and reflect.

Image Trace would get you . . . somewhere . . . but if it was my task, I wouldn't even consider it. I'd expect I could (re)produce this manually in a couple hours or less. Often, in the time we take to find an "easy button," we could be halfway to accomplishing something the hard (and better) way.

That depends on how the shirts will be printed and what color fabric.  There are inkjet shirt printers that can reproduce that image.  Not sure Illustrator is the right application to recreate that image.  You can Use Illustrator to build the file, but for the image' continuous tone, you really should consider the print device and its ability to reproduce.  If you print on a dark fabric, you will need White ink.  A shirt printed in silkscreen probably is not ideal, but possible.  The silk screen itself is a challenge.

You can try using Illustrator's Image Tracing feature. Open the jpeg in Illustrator and select it. The Control Panel at the top of the screen below the Menu (which works contextually) will display the Image Tracing feature. Use the pull-down menu right next to the words Image Tracing and choose a style that suits your need and the tracing will begin. When the tracing is finished click the Expand button to make it into live Illustrator art. In the screen shot below I took your image and used the High-Fidelity Photo option (middle image). The image on the right shows the result in the outline mode. A word of warning though this is a very imperfect feature. The finished art is a mass of compound paths that can be quite difficult to wade through and work with. Occasionally though it is the right feature to use. For more information see https://helpx.adobe.com/illustrator/using/image-trace.html.