I wanted to make this as a buzz on d.buzz, but I couldn't manage to get it concise enough. So this is my little showerthought with too many words.
If you think about it SBD is debt against Steem holders (both vested and liquid), indeed it was designed and described as such in the Steem Whitepaper.
In the startup world, convertible notes are short-term debt instruments that can be converted to
ownership at a rate determined in the future, typically during a future funding round. A blockchain based
token can be viewed as ownership in the community whereas a convertible note can be viewed as a debt
denominated in any other commodity or currency. The terms of the convertible note allow the holder to
convert to the backing token with a minimum notice at the fair market price of the token.
As debt on Steem - taking on the full burden of SBD was not something Hive had to do in the fork, as SBD holders would still be expected to get ~$1 returned to them on the Steem branch of the fork. I don't know what the rationale for Hive taking on the HBD debt was, but I think nonetheless it demonstrates legitimacy of continuity in that the debt was held against the community, not simply a blockchain. With the community moving to a new chain, so too should the debt held against that community.
It demonstrates legitimacy in another sense too, because with the debt burden comes the risk of debt overburden and potentially extreme levels of inflation where the new branch cannot sustain the debt. Although Hive did experience higher conversion-caused inflation than Steem after the fork, it was not extreme or catastrophic, and it has managed to recover and reduce inflation since.
If you look at other forks of Steem, for example Blurt, it does not appear that they have taken on the debt burden with it. Perhaps for some both the reasons above - the debt burden would be dangerously high, but also somewhat an acknowledgement that the community did not come along with them in the fork.
The inverse would also be true - if Steem decided to remove the SBD due to the heavy debt burden, it would be a sign of the illegitimacy of that project.