The USS Gerald Ford will stop to render assistance to those in distress at sea even i it wasn't it's primary mission, just like the USS Enterprise has.
Yes, however...
The USS Gerald Ford will offer aid in a case of a humanitarian crisis even those weren't its captain's orders when they left port, just like the USS Enterprise has.
Not without orders from the Pacific Fleet, no. As a military vessel, its movements must be coordinated with the overall defensive strategy of the entire armed forces in mind, and the captain of a carrier battlegroup is in no position to make those kinds of decisions. Even the rendering of situational aid to vessels in distress cannot be interpreted as a legitimate reason to pull the ship and/or the entire group out of position.
So if a tsunami hits American Samoa a hundred thousand people are suddenly without power, the Captain radios the Pentagon and asks for instructions. They might tell him "Render any aid you can and more help is on the way" or they might tell him to stand by and monitor the situation as best they can. Or they might tell him "Get the hell out of there!" because they know something he doesn't. About the only thing that WON'T happen is the Captain deciding entirely on his own to commit his entire battlegroup to humanitarian relief just because he can.
It won't do it in a time of war just like the USS Enterprise won't.
You mean like the Enterprise DID in "Insurrection?"
Or, for that matter, like the Enterprise-D did in throughout Seasons 1 and 2 when the border wars with the Cardassians were still very much in progress? Let that sink in for a minute: how much of a "military organization" could they really be if half the fleet is off fighting a war and the OTHER half of the fleet barely mentions it at all? The Federation was still in conflict with the Cardassians at the moment Will Riker said the words "Combat... is a minor province in the makeup of a starship commander."
It may not fit your definition of the 21st century military
And it doesn't fit
their definition of a 24th century military. That's been stated canonically and isn't up for debate. The only thing left to debate is "Why?"
If you think their definition is wrong and Starfleet officers are just a bunch of liars, then we have even less reason to believe them when they say "I'm a soldier" or "I'm a doctor, not a bricklayer!" You either try to make sense of what they say, or you just make shit up to fit your personal preferences.
Probably because that never happened! What "exploration mission were conducted by knights" on shining armor and "on horseback"???
Vasco de Gama and Pedro Cabral, just off the top of my head. One or both of them were what Q was trying to imitate in "Encounter at Farpoint" when he appeared in plate armor with a fuzzy hat, and is essentially what he was trying to accuse Picard of being the descendant of.
Before them, the various Crusader expeditions into the middle east were essentially military expeditions with knights as their principal soldiers and explorers aiding the discovery of new routes and new resources during their travels. But if you look back to before the "Age of Discovery" from around the 15th century onwards, you'll find that most of the European exploration of the world was NOT actually conducted by the military, but by Franciscan and Dominican missionaries in concert with merchants and traders.
So we can add that to the list: Picard is not a military commander, he's not a knight, he's not a merchant, and despite his bald head and high-minded morality, he's definitely not a monk. And all in all, historical comparisons to the organizations that IN THE PAST once filled Starfleet's roles are irrelevant: Starfleet is none of those things, Starfleet is Starfleet.
