If I am speaking to a client, no way am I answering my phone to speak to another client.
If I was that client, I would be looking for new representation after that meeting. If the phone call proceeded to take a long amount of time.
What your boss should be looking at is how fast those voice mails are answered. If they are answered promptly, I see no problem for a call to go to voice mail.
Read your update. I don't mind if my vet's office has to answer the phone, but if I am standing there and it is not an emergency, than I expect that call to be put on hold until they are finished with me. And honestly, what vet's office is not busy? My vet's office is always busy, and I have no problem with that. I know it. I expect a few phone calls while I am standing there, but I don't expect every phone call to take a lot of time either.
If your boss wants all phone calls answered no matter what, then maybe he should hire someone to specifically just answer the phones in an area away from in coming clients.