This is a really nice idea. This already exists for some companies in the form of Twitter accounts. I doubt the people doing the typee-typee actually has any authority over business processes, or demand a change, but I think they at least have the business owners on their speed dial, vs normal support tickets. But having an email is far better.
Amusingly, Twitter itself could use one. My account (@acangiano) has all of its images censored under "sensitive content" even though they are 100% benign images. No amount of tweets to @TwitterSupport has done anything at all to change it. There is basically no recourse. My account is like 12 years old and has 4.5K followers, so it's not like it's a random spam account, either. ¯\_(ツ)_/¯
I find this whole infrastructure, despite my taking advantage of it, to be very flawed. There're many customers who might have alarming issues, who never get attention because they're fearful to be perceived as a "Karen" for bitching on Twitter.
Yes it's a good point that companies currently have some sort of Twitter presence trying to address bad publicity posts, often they seem to be able to get things done.
The Ombudsman role is there to get things fixed when all else has failed, and before the angry customer posts to social media.