So I'll assume the arguments that you skipped you conceited?
What arguments though? Buggy having found fortune, which may or may not resulted in him having huge military might? Well, that's up in the air. Zoro being embarrassed that his end goal serves Buggy? Faulty too. Zoro's end goal is to be the most powerful swordsman. Mihawk serving someone else or being killed off even would not change that. Zoro doesn't even give a fuck about Mihawk on a personal level, tells him to fuck off:
So why would you argue Zoro only talked highly of Mihawk to secretly save face? That has absolutely no basis and quite frankly, it's not even my job to disprove that, as it is your job to prove the claims you make rather than just make it sound like it's "obvious".
No, because Zoro and Sanji are not rivals, they are just being competitive. Shanks and Mihawk however had battles which even Whitebeard remembers among the legends of the past. With secondary sources saying that Shanks was the one getting famous through battling Mihawk.
Because according to you he's that strong? What, couple fodder ships are too much for him? That's because Whitebeard was sailing on a ship and had a crew with him. He did not have an island like Mihawk to chill in one place, he was sailing.
So the fact Whitebeard was sailing means he cannot fight those marines off? It doesn't matter how strong Mihawk is, as the Marines would keep sending forces endlessly, even admirals at some point. Mihawk chose that location to drink wine and read the news. Obviously it wouldn't make sense to stay there with the entire marines targetting the island.
The last time we saw him he was literally getting excited about being chased. So what happened? Now he has to team up with others for protection...
Yeah, the fact he was getting excited proves that Mihawk needs no protection. Him joining this organisation out of the necessity of being protected is another headcanon you pulled out of your ass. Prove your words, cite where it was said Mihawk joined for protection. Well, you can't, because obviously, you are talking rubbish. You saw a pic of Mihawk being depicted along with Crocodile and Buggy and somehow fabricated the theory Mihawk serves under Buggy and needs protection, even through everyone - Shanks, Luffy, Zoro call it bullshit and Buggy a fraud.
Are you even interested in facts, or are you just wasting my time with your cope?
Did he not apologize to Shanks and say that he won't hold back?
Not holding back =/= going all out. Kaido too told Yamato that he won't hold back. Last time I checked, he didn't use his strongest modes (Shuron Hakke, Flame Dragon Torch). Holding back is any attempt to not harm the opposite person. But as Mihawk was testing whether fate will protect Luffy, he obviously wouldn't restrain himself.
But to assume that Mihawk went all out on a pre skip rookie Luffy is pretty silly, isn't it? Zoro obliterated King (YC1) and still remembers his promise of facing Mihawk, which won't happen very soon. And you think Zoro's end boss can't beat pre skip Luffy when going all out.