Use the original Leica IIIg case
Mr Zhou and Luigi make cases for the IIIg. Unfortunately, their case designs block the self timer. I wrote to them about it, and their response was essentially "take it or leave it, we're not changing anything".
I do use the self timer on my cameras, and those that don't have them I have the Knips and Kopil clockwork timers.
Anyway, I have one of Luigi's M camera cases that wouldn't allow me to mount a goggled lens on my M4-2. That was the last straw. I took a good sharp hunting knife on the kitchen cutting board, and cut off the offending bits of leather on the case. Now I can mount lenses without the case interfering. The raw leather edge I dressed up with Fiebing's Edge Kote.
I had to do similar case surgery on a Luigi case for my Canon 7s. The way he made it, the fit was poor on the camera.
So, I'm done with Leicatime cases. For my M cameras, I love the Artisan and Artist cases, which give you clearance for the self timer.
Back to your question: the original Leica cases are the best for the IIIg. I cut off the original leather straps, which are dried out and useless by now, and use a Nikon AN-4B strap. The Nikon strap has no visible logos, it is black in color, and of soft woven nylon. It will never break.