Factors that determine whether a character with a gigantic ego will become loved or hated for it
Characters who brag about how awesome they are can be split into two groups: Those who have the competence to back it up, and those who don't.
Justified arrogance is a useful trait when writing villains. It can make them more threatening. It can make their evilness more enjoyable. It can make the hero's eventual triumph that much sweeter. When it comes to heroic characters, justified arrogance can make them more realistic. Making a highly competent hero too humble could come across as unbelievable, as if they're trying too hard to be a goody two-shoes — or even trying to mislead others about their true power. In escapist fiction, the arrogance of a hero might be something the viewers wish they could get away with in real life. There's also the fact that a hero being arrogant towards their vanquished foes can simply be more fun than them being humble in victory — especially if the foes are Hate Sinks who deserve a dose of mockery.
On the other hand, an arrogant character with too much power will likely grate on the audience's nerves. Invincible Heroes and Invincible Villains are already at risk of Scrappyhood because they're often perceived as overpowered. If they constantly rub it in, it'll just exacerbate the hate towards them.
A weak and/or incompetent character whose arrogance is entirely unjustified can still end up becoming popular if their ravings are more entertaining than annoying. However, this is risky: even if the raving themselves are funny, it's easy to have the character's actions render them unlikable. If the incompetent character is useless, but also harmless and doesn't overstay their welcome, the audience might find them endearing. On the other hand, if the arrogant character is an annoying Load who always needs the heroes to take care of them without contributing anything useful in return, their arrogance will just make them more annoying. Even worse: if the character is a Millstone who's actively making things worse for the heroes, the audience will probably find them anything but endearing — especially if their arrogance is the reason they're a Leeroy Jenkins who constantly ruins their plans.
