It is easy to set up Gmail such that Google doesn't have access to the text content of your mail -- simply install a desktop mail client (e.g. Thunderbird) and enable encryption. But now you've essentially disabled every feature of Gmail except for being a dumb SMTP relay.
The truth is, when you contact someone, you don't know what is happening to that information. Gmail, if anything, makes it more clear than most other means of communication.
Assuming there were some terms and conditions on who could access those letters, sure! In fact, if the additional benefit was they would weed out all the junk mail I get to only provide me things I'd be interested in, that's even better.
Of course, this also makes the assumption that their are still other means of communication (like phone). If the only means of communicating with me are through this service, then no.
> Do you think your correspondents would be OK with it if they knew you had such arrangement?
My correspondents? Yes. Yes I do. In fact, I believe most of them would want a service like that for themselves.
For things people wanted to send privately, they could pay to have it sent to me, like they can now.
So yes, I don't see a problem.
Would you be OK with letting a company to scan and retain a copy of every letter you sent and receive in exchange for being able to not use stamps? Bonus question - do you think your correspondents would be OK with it if they knew you had such arrangement?