I guess it doesn't matter that much, but if there were a common language, it would be a nice feature that would allow you to skip the middleman altogether and have simpler dependencies if you know which solvers you want.

It's the common language that matter more though. I've used GAMS in my undergrad, so I'm under the impression that it's had at least some adoption, but it's proprietary AFAIK. I've know some people who use AMPL too, which also seems popular, but I'm not sure about it's licenses.