This is a really peculiar concept to me. They're just emails. It's meant to be a quick and efficient form of communication, not a bureaucratic nightmare! By the sounds of it they turn it into a five-man operation or similar. How ridiculous.

I have never emailed my private therapist. I know I could, though, as she has her own private practice and would simply read, and maybe respond, without any further hassle. And I used to email my uni therapist all the time, and even though she is employed by the university there was no craziness to it; I sent the emails to her private email address, she read and responded to them, and that was all that happened. She communicated via email and text (and sometimes phone calls) between sessions because she wanted to, not as part of her job (I know this because she told me).

I would never send an email if I knew it would end up going through a process like the one you are describing!