I never said Lindsay was a conservative Republican - I said he was a Republican. Yes, of course, Koch won the Upper East Side. He was a very popular conservative Democrat. I voted for him in the general 3 times.
Koch was a lot of things, but he was a pretty doctrinaire leftist in his 9 years in Congress, earning a 96% left-wing (or 4% Conservative) lifetime rating from the ACU (almost identical to Bella Abzug’s rating). When he was Mayor, he clearly lifted the spirits of the city after 12 years of rank incompetence under Lindsay and the hapless Beame, but he never really was successful with tackling the underlying issues that had been festering since the 1960s (namely the awful crime rates), and by the time he was beaten by the even more execrable Dinkins, the murder rate was at or above 2,000 per year. A lot of flash, warmth and panache, but not a lot of substantive accomplishment (which didn’t come until Giuliani).