The Man With No Name didn't actually kill Evil Eb's Littlest Brother, Billy Bob did, and the Mistaken Identity plot goes even deeper than you think.
It really does make no sense for Eb to be so friendly towards the guy who killed all (save one missing content exception) of his close ones. However, in line with its parodying of Western tropes, perhaps the cigarette man doesn't live up to his image outside of his shooting skills, and kicked off the whole plot by making a dumb mistake. This coincidentally lines up with Billy Bob killing Eb's real brother elsewhere, and due to the cigarette man's misinformation, the gang attack the Man with No Name. The reason Eb is so forgiving is that the man's done nothing wrong, and he needs to rebuild his gang anyways. Well, besides pulping that poor spider.The Hole-In-The-Head Gang is actually comprised of a few Evil Brits who migrated to the US in hopes they can get rich.
Aside from those who do speak with an American accent, the Hole-In-The-Head Gang is an Equal-Opportunity Evil group, judging as how mostly everyone speaks with a thick British accent (most notably Wildcard McVee, and McVee itself sounds like a British or more specifically Irish surname). While there are those who speak with different accents (like Nasty Ned, who speaks in a slight Mexican accent), the group is mostly British led by an American man (Evil Eb). It could also explain why some of the voice actors were Not Even Bothering with the Accent.