As to whether SF always viewed it as a symbol of White Supremacy going all the way back to the war, I don't recall that, and I'm a lifelong Northern California resident. If that's the case, how did it ever get put up in the first place, much less remain flying during Moscone's years?
I don't know whether you meant this as a serious question or just a rhetorical one, but I never miss an opportunity dive into the history of my favorite American city, so:
Short answer is that the person responsible for putting it up probably wasn't thinking about the racist implications of doing so
per se, but the City most certainly took umbrage, and almost immediately after it started flying. The concept of the patron, a local business magnate and conservative lobbyist was to fill some then-empty flagpoles with a historical tribute to 18 different American flags. The second they all went up, the Confederate flag was definitely noticed, and there were protests in the street by the end of the day. Racial tensions were sky high in 1964, and many of the Confederate flags lately taken down around the country were first raised that very year. By the end of 48 hours, the flag had been taken down and vandalized, a much more audacious act by early 1960s standards than it would seem today. As now, it became a fracas, with enraged citizens on both sides of the issue and embattled politicians trying to find "compromise", which they did by replacing the Battle Flag with the lesser known Stars and Bars, seemingly hoping that people would be less likely to recognize it and take umbrage. The ACLU took thew side of the flag-fliers, again as now, and they had considerable pull in 1964. But it didn't actually put the conflict to rest. People continued to complain on the regular, and caused a major uproar in 1981 when the flags were all replaced... and the Battle Flag was once again included in the set. Protests once again began that very day, and continued for the next few years, temporarily ending with the City Council taking the side of the flag-fliers. However, this was to end in 1984, on the twenty year anniversary of the original erection of the flag.
The idea that SF is a purely liberal town with only liberals in it, promoting liberal policies with liberal ideas and never any opposition is an invention of the conservative news media. The City of SF may have a liberal majority, but that doesn't mean centrist values never prevail, especially on race. No one would look at Fog City even today and go "Wow, what an exemplar of racial equity and integration!", let alone as things stood in 1984. Many neighborhoods were effectively segregated. Schools likewise. Even today, English has official status and bilingual education is banned. Affirmative action of any kind is banned. Redlining was prominent until recently, and still quietly and informally occurs under the aegis of gentrification and "urban renewal".
But the brutal murder of Moscone and Milk pushed the city much further left, calling conservatives on their bluff of feigning "unity" while also fomenting political violence, in a situation we see echoes of today. History does not repeat itself, but it rhymes. When the flag was once again vandalized and dismantled, Feinstein again tried to defend the status quo and put it back up. But the political tide had changed, there was now a black woman on the city council (Doris Ward, the city's first) and Feinstein was a less experienced politician than your average San Francisco mayor. After several attempts to replace the flag, each ending in a new and more extreme act of vandalism, she caved. I don't think she would have changed her mind on this if it hadn't been for a full twenty years of citizen action trying to get the flag removed, so I do not see the concession as some sort of redemption arc. She likely still doesn't "really see the problem" with the flag, like a lot of wealthy Americans not of a background to be particularly hurt by the flying of a symbol of hate. Cetainly she did not make any real vocal concessions at the time. She was quite clear in her statement to the press:
“I want to make it very clear that my decision is based only on Supervisor Ward’s request,” Feinstein said. “I’m not impressed because some group shinnies up a flag pole and tears down a flag.”
Yeah, that's not exactly a change of heart.