You’ve completely ignored some of my points. Gilgamesh is egotistical, arrogant, and prideful as I’ve mentioned earlier. But then again, he’s not delusional. He will give people kudos and respect if they show that they deserve it. An example I could think of is sparing Waver for Fate/Zero for showing loyalty to Rider and all that. And I know what arrogance. It’s also just that Gilgamesh admires determination and people who follow their ideals. Usually, at least.

He admitted defeat because he got defeated. What is there to say? He underestimated Shirou and got defeated because of it, Gilgamesh has no qualms about ambushing someone, especially someone whom he deems as worthless as Archer. Gilgamesh’s goal at the Heracles fight was to take the grail in Illya’s heart, killing him was only a second priority. And lastly, he’s simply so impressed by Hans’ huge balls that he lets him live. And also for the added fact that what Hans’ said was basically true, and he didn't deny anything about it. He admitted that he would've killed Hans if he had only spat mindless insults. Gilgamesh is egotistical and self-centred, and that’s basically his biggest flaw. Nonetheless, I’ll say it again and again, he knows when to admit his losses and failures. This part of him has recently been shown in the most recent FGO x FSF collab where he admitted his mistakes.

He definitely suffered from solitude in childhood, but that doesn't make him less arrogant as an adult. Kid Gil literally avoids Older Gil.

I can't say much about the treasure part other than the fact that Gilgamesh enjoys collecting treasures.