Of course names of the experts will be published! Names, social accounts (if they want), descriptions.
Users can search for an expert using a filter or simply by name.
I think that many qualified specialists will not agree to openly work as experts. Many are already bound by obligations or contracts with their current employers. In parallel, officially cooperating with your platform, they will violate the terms of their contract. Therefore, I think you need to consider a variant with anonymous experts.
Anonymity is excluded. Users have the right to know who they are communicating with.
Nobody forbids people who work under a contract to communicate in social networks, to help other people solve their problems.
This is not a job for experts. Services are paid only to ensure that users think who they really need to write. And not spamming everyone. I think even the most popular experts will receive no more than 10 emails per day. But experts would receive several hundred letters if the service were free. This is the essence.
P.S. We call "experts" not only business professionals, but also investors, advisors, mentors. These people receive an ocean of letters daily. It's impossible to read everything, and really interesting sentences just remain unread.