Ok I want to break this down piece by piece
Yes he swore to avenge Yasu, but his vengeance was directed at Orochi. Kaido had nothing to do with Yasu's death.
I'm not entirely sure what the second half of this quote means, but he gave up one valuable sword for another. I guess it's a big deal, but this sword still made him much stronger than he was before, so I don't think it really matters.
Yeah, I'm fine with him getting up and fighting. Hell, let him fight and beat King I don't care. I just don't see why he needs to kill Kaido.
I dont see why beating King isn't good enough or "wasting time". King is still a Yonko 1st Commander. Same general level as Katakuri, how is fighting somebody on that level a waste of time.
1. He swore revenge for Yasuie but doesn't possess knowledge or info of Orochi. Before and after the "death" on stage, Zoro's target was Kaido. We saw that in both encounters with Apoo. Even if someone else did put Orochi down for good, it doesn't matter because that has nothing to do with what Zoro himself swore. But back to Zoro seeking Kaido, because Orochi is a complete worm who would have not been anywhere close to killing Yasuie without Kaido backing him... unless you think Oda is building up to a neg diff of like, a Brook level villain.
2. The understanding of Shusui in Wano is first that Zoro was not chosen by Shusui, that it does not belong to him. The second is that Zoro's tie to Ryuma isn't a real thing. This goes to Kawamatsu believing that he is delusional. Modest achievement means upholding these understandings, which aren't true. At the least Zoro will cut down King but we'll get to him in a second.
3. Zoro is on a time crunch. Once the elixir wears off he will feel Indra, Hakai, and Thunder Bagua multiplied. He was able to injure Kaido when he was barely able to stand, so why would he go for King? "He already knows Luffy will win." He thinks Luffy will beat all these guys eventually but that didn't stop him from going up in the first place.
4. I didn't forget about King, but you must have forgotten about Marco. This guy has a healing factor and aerial combat abilities that make him perfect for King. But King knocks this guy out and Zoro picks up the tab? It's possible but it's pretty dumb if the guy who put Zoro on the roof in the first place actually needs him to beat King. Especially since Marco was able to hold his own 2 on 1.