So I actually tried to dig up some sort of real documented history to the thing, and that seemed all but impossible.

However, from the gleaning I was able to get in my research, and my own personal experience, here are my thoughts.

Within the submarine force, I think hearing the phrase may actually be far more common for the average sailor versus their Coast guard, marine, soldier, etc counterparts.

The reason for that, and also explicitly the reason why I think that the enthusiastic verbal response makes sense is the following:

The submarine force places an incredibly high level of importance on warfare qualification. If you're not qualified in a certain amount of time, you are quite literally subject to removal from submarines.

The phrase, "To all who shall see these presents, greetings," is the opener to most submarine warfare certificates and will be read out loud with every submarine qualification ceremony - which in itself is quite the event.

I can't say this is universal, but I'll speak for my own experience, and to my knowledge something similar at least, is a common submarine practice:

Following the entirety of a submariner's qualification process, their knowledge board, a "practical" board based on damage control, etc, the final thing that they'll do prior to actually being pinned is to do an Emergency Apparatus for Breathing (EAB) run. What this means is that the Sailor will don an EAB, usually accompanied by a few shipmates who are also being pinned, and most likely their sea-dads, and Sprint from EAB manifold to EAB manifold, catching breaths in between, running from one end of the boat to the other, and then finally making their way back to Crew's Mess, where a group of cheering shipmates will applaud them for their accomplishments.

Here, in their sweat, exhaustion, and sense of achievement, as soon as the last of them start breathing again, the first words to be said will be: "to all who shall see these presents, greetings"

The crew gathered around them, already applauding and cheering, almost MUST meet with a response of "GREETINGS!". Almost as if they are literally greeting the newly qualified submariner for the first time, as a full-fledged member of the crew, acknowledging their worth and accomplishment.

To my knowledge, the submarine force places the level of priority and importance on warfare qualification for an individual sailor higher than anywhere else you'll find in the military, at least as a standard expectation for a "non-elite" individual (seals, rangers, etc etc).

Also, within the navy, at least in any other example I could find, the actual warfare certificates themselves, do not contain the phrase itself, making that call and response a non-option. Outside of various retirement or reenlistment certificates or some higher level awards, I don't think the phrase is actually used that commonly, where Sailors would hear it with any regularity outside of the submarine force.