The US government does have storage facilities and secure messaging tools with escrow, all designed for exactly this use-case (secure messaging amongst DoD personnel.) But the whole point of Signal+TeleMessage was to route around that "clunky stuff" by outsourcing it to a vendor.
I understand the point - the point is people who believe the size of their bank account is proportional to their intelligence and aptitude at everything making flat dumb decisions because, if anything, the relationship is none of not inversely proportional. Their arrogance and eschewing of expertise in favor of magical thinking will end up with a lot of people dead.
The DoD obviously has a need to message with people who don't have access to their hardware. Signal can basically do this on its own if you link a Signal account to an Internet-connected PC and back up those messages, I don't see why you would want a third party app involved.
It seems likely to me that this was the "whole point of Signal+TeleMessage" and then in addition to being a bad solution, it got misused for communications that shouldn't have left the DoD's networks anyway.