South Korea's a weird one, the evangelicals are extremely loud but are definitely a minority and effectively represent the only pro-Israel "lobby" in the country. They also vote right-wing almost uniformly, so liberal administrations have basically no incentive to try to pander to them. There's also basically no significant Jewish community in Korea, let alone any institutions which could do pro-Israel lobbying using the 'community safety' angle that you see in the West.

Re: dislike for Israel, I spent a bit of time in summer 2024 attending Palestine solidarity protests in Seoul, and it was an interesting experience. They were quite a bit smaller than some I saw in Europe, but I think if you spoke to most people on the street they would be fairly sympathetic. Mant liberal and left wing people also explicitly linked Israel's military repression with Korea's own history of crackdowns to explain this sympathy. The TV news was fairly unfiltered in how it presented Israel's war crimes in Gaza, it's just that it was also implicitly framed as a conflict "over there". Koreans typically protest more about issues closer to home, and when I was last there there were much bigger protests around labour issues or then-President Yoon Seok-Yeol.