I wonder if it would be possible to allow people to ssh into the edge servers with enough access to verify no access logs are stored but not enough to cause any problems. Admit i have not thought it through but would be cool having people verify the live environment while running.
You can't really verify anything in this way. SSH is just a protocol, you're trusting the SSH server to give you a shell inside the real production environment instead of giving you a shell inside some elaborate simulation of a production environment. It's about as trustworthy as a policy page saying "we don't keep logs".
You are correct. Would need something like distributed ledger to fully prove things.
It might not be possible to verify 100% but the more transparency the better i guess. Seeing the 3 way handshake and connection information, the timings, location of the server. Would need to be quite elaborate to fake. Just thought was a fun idea. Have the customer allowed in to production. A lot more difficult then publish privacy page, source code, fake audit reports.