That's a good argument for using PGP/GPG/some other secure communication protocol.
There was one email provider that didn't, remember what happened to it?
And having your bank share your lawyer with you might potentially open up conflict of interest questions, since your legal rights and your banks might diverge.
In-house council for a company get away with this (for internal communications) because they are lawyers for the company (right up until your interests diverge from that of the company, at which point you really want your own lawyer), but for your bank this stops being true.
Basically, you are thinking about law like an engineer, not a lawyer.
A third party can be necessary to facilitate the communication between the client and the attorney. A clearer example was if you needed an interpreter to speak to your lawyer - surely that conversation remains privileged and the third party in the room cannot disclose what was learned?
Of course.
The technical "gotchas" people are coming up with here do not stand in the real world