Everyone has to make their own risk assessment, but I have no concerns of ozone on my saddle, bridge or tether.
I've been using O3 in totes to treat gear for over 10 years and I've see no degradation of anything other than elastic and rubber, and even at that, the damage has been very minimal. It could be because of how I use O3?? IDK.
Everything I treat does not get stored in O3, the O3 is only used for short treatment times to deodorize and once the cycle is ran, all gear containing rubber/elastic is allowed to off-gas, and then stored in tubs or bags to protect from contamination.
The other garments that don't contain rubber/elastic may or may not be allowed off-gas. Some of my clothes remains in an O3 environment for more extended periods and I believe, if O3 was harmful to non-rubber gear, I would be seeing major damage. A lot of my clothes have been repeatedly treated for more than 10 years with no signs of damage.
Rubber boots are another issue. O3 does reduce the life of boots. Not horribly IMO, but it does shorten their life. I still treat boots as I feel boots are one of the main odor producers. I'm willing to buy a new pair of rubber boots every 3 years if it means reducing odor. It's a small price to pay.
But O3 treatments on life supporting gear is an assessment that each and everyone has to make for themselves.