[mrsmith]

I think it can be kept safe just as software certificates have.
Warrants are low barriers but they’re all we’ve had for hundreds of year.
People would, of course, still have their own right to challenge the warrant.

I picture the basic service like a bank’s for deposit boxes:
“To summarize, the search warrant must specify who (box renter, issuing Justice), when (day or night), where (which financial organization and if possible which box) and why (probable cause). “
The ‘deluxe’ service could go all the way to putting a duty on the encryptor to act as attorney for the client.
I won’t guess on marketability of the arrangements other than to say it’s ‘possible’.

Reading about safe deposit boxes brought up a totally off-topic benefit to the system: allowing access to the data in the case of the death or incapacity of the owner.

[Swordmaker]

“To summarize, the search warrant must specify who (box renter, issuing Justice), when (day or night), where (which financial organization and if possible which box) and why (probable cause). “

Bank deposit boxes are broken into all the time. Even our most secure banks have had secure deposits stolen from them.