The first words Flynn says to Gil are "My King". Not "King Gilgamesh""Your highness/majesty/goldyness". Flynn put himself in a subordinate position immediately when trying to butter Gil up.
Then the correct option is to acquire said apple. Or from Gil's prospective, Uruk has a broken leg and needs a split but Flynn wants to apply CPR. So Flynn left to find someone who wanted CPR, leaving Uruk with a broken leg. Flynn wanted to help but only in the way he deemed appropriate which Gil sort of addresses: Who is this guy to know how best to help me, my city, my people and my kingdom? And Flynn's actions proved Gil correct.
Gil gave Flynn a chance and Flynn said 'Fuck you' and burned that bridge. He could have gone to Gil and ask 'What the fuck?', he could have asked Siduri if Gil said anything else, he could have used the opportunity to gather intel of the situation from the people first hand. But no, at the first sign of inconvenience, he left and went right back to his old methods.
On a similar note, Flynn won't shear sheep because Gil won't explain to him why but he'll poison Chaldean staff because Medea did? That does not paint a good picture of the guy's character.
Gil most likely wouldn't have accepted anything less than people calling him 'My King'.
As far as he's concerned he's the king of everybody and if they don't address him like that then they deserve to be killed.
Yeah Caster wouldn't do that, but Archer would, and Charlie has so far dealt with the Archer version the previous times he met Gil.
And that's another thing, the previous meetings with Archer Gil.
Archer Gil is someone that would consider himself to be the only one that matters and can solve all problems by himself, so he'll send other people, even if they're extremely competent, to do useless work that helps no one.
Sheering sheep doesn't stop the monsters attacking the city, it doesn't stop the goddesses.
A bunch of powerful Servants however do stop those things, but not if they're spending time doing farm work.
Charlie has so far dealt with only Archer Gil so he has some bias against Gilgamesh, and when he sent them to do useless work his prejudices against him were just confirmed.
Charlie didn't offer to do CPR so much as he offered to give Gil and Uruk the Super Soldier Serum, and yet Gil didn't take it, despite the very clear benefits, so Charlie decided to go to someone that would take it instead of wasting his time.
As far as Charlie is concerned they don't have time to waste on stupid crap, like what he sees the mission Gil gave them as being, because the world is on the line.
If Gil told them to retrieve some sort of weapon or kill some monster then Flynn may be more cooperative, because those are missions that make sense in proving that you can help defend a city.
Sheering shetp proves you can jyst sheer shetp, that's it.
Medea was also an ally and has proven to be competent.
Charlie also could have changed since his poisoning allies days.
People do change.
