ASSRA rules forbid modern shooting coats. That is new to me. 
Generally, Coors matches and ASSRA/ISSA matches held in the Western US do not allow modern shooting coats. 

Both times I shot at ASSRA Etna Green matches, modern shooting coats were allowed and worn.
Maybe the rule was changed and I'm not aware of it.

This is a Schuetzen Match in progress at Shellmound Park long ago, courtesy of Randy Wright.

In ISSA matches all one has to do is wear a big honkin winter coat to cover up your preferred girdle or elbow rest. 

   

KaiserKong
You asked about shooting coats back in the day; in reading original sources such as Shooting and Fishing magazine, there were a number of references to limiting clothing to support the shooter or rifle. I've seen passages that it was not allowed to tuck the butt stock under suspender straps. Also, it was not allowed for a shooter to button up their coat and place the butt stock into the coat.
I'm sure then (as now) no one ever cheated. It's a gentleman's game after all 

Thanks Randy and Frnkeore for the bit of history. 

I do have a modern shooting jacket that I wear when firing modern free rifles and can feel it helps minimize the muzzle some sway compared to without. So it is definitely it would be an advantage for Schuetzen shooting as well. I haven't had a chance to make it out to any ASSRA competitions - so many folks wear modern jackets? Perhaps it could be a separate event? 

I also recently bought a Bavarian/Austrian style wool coat similar to what would have been used at the time my Aydt was popular. Even though it's not stiff at all I do notice the friction from the wool does help my left arm 'stick' to my chest/hip better. So there probably is an ever so slight advantage there. But as noted earlier the German rules made wearing a coat compulsory so that would level the playing field. Similarly I've read in the German rules that putting the stock inside the coat was forbidden. 

-Dave