Segregation is a broad term that covered many different things and if you say Biden "supported segregation" you can be taken to mean "Joe Biden approved of separate drinking fountains/restrooms/restaurants/etc. etc. etc. for whites and blacks," which would be ridiculous.
Specifically, some four decades ago Biden was a critic of court-ordered (aka "forced") busing of students to schools outside their own districts as a means of integrating education. That's the issue that Kamala Harris brought up in the debates; she was bused as a child and sees herself as having benefited from the program. Busing was hugely controversial in its time and there were all sorts of arguments pro and con that don't fit neatly in a simple "racist vs antiracist" binary. (It also included class elements: One big complaint in Massachusetts was that the program moved poor kids around the school districts of Boston while leaving the more affluent white kids of the suburbs untouched, prompting charges that lots of white suburban liberals were happy to impose such programs on poor and working-class urban residents so long as they didn't have to abide by them themselves.)
Regardless, please note that Biden was critical of busing rather than of social integration generally.
I'm really sorry, but we all understand what this means, and this comes across as needless apologism for Biden.
Very few supporters of racism, or segregation deal with the core issue at hand. They find bits around the edges that are more palatable to chip away at. You see it with the anti-abortion lot here too. They know they can't win the debate on abortion, so they chip away at term limits, bit by bit. They have the same intention (to end abortion) but there's a dishonesty to their objective as it's cloaked.
Brown V Board of Education was a key moment in US history, and a key moment in the dismantling of segregationism. Schooling was of course a very important and significant element of that (as much as water fountains, or toilets etc). If you side with the segregationists on the matter, essentially by finding a convenient loophole and giving political cover to their arguments, you are every bit as responsible for it as they are. In many ways you are more respnsible, as there is a dishonesty to your conduct.
Lets not whitewash people's behaviour here. There is a BLM campaign, that is rightly pointing out how deep racism runs, and ultimately how structural it is. Pretending that certain white men who played important roles in upholding the system should be absconded from this, on the basis of them either not really liking it, or not really understanding it is an insult to the black people who suffered under that system.
As he said himself, he had to work with the racist segregationists to get things done. Thats fine. But he has to own his decision and the understandable flak that comes with it now that he helped to soften and legitimise with cover about busing being unfair. It's nonsense.