I'd be really interested in hearing from some minority - whether in terms of race or sexual orientation or whatever - who wouldn't want her community to accept a coin toss on the principles described above.

I’m white and I wouldn’t be favourable towards the coin toss idea, but I’ll answer anyway since some of the reasoning might be the same.

Suppose that we had a machine that could change people's skin color, physical features, and speech patterns, so that people of one race could be turned indistinguishable from people of another. And suppose we wanted to end all racial discrimination forever by making everyone the same race. So we flip a coin to decide whether all white people have to change into black people, or all black people have to change into white people - discrimination disappears either way, and this way we know it's not a power thing where white people are trying to enforce their own norms.

Firstly, changing surface appearances wouldn’t necessarily end racial discrimination (i.e. physical similarity doesn't guarantee the absence of tribal identification; discrimination may be based on alleged biological differences that are not limited to surface characteristics). Furthermore I don’t see how pair bonding and attraction, and personal identity could be adequately preserved (for people in general) through substantial changes in physical appearance. For the sake of the thought experiment I suppose we can ignore this, though.

I would nonetheless object to the idea on the basis that it is needlessly illiberal. Why not just allow anyone to use the machine if they want to do so? If someone feels he is being discriminated against, then he is free to use the machine. If he is unwilling to use the machine, presumably the problem isn’t bad enough to merit trampling over the personal liberty and aesthetic values of others.

I'm white, and I don't think I would object too much if the coin came up as "all whites have to change to blacks." I could see some white people objecting on aesthetic grounds, that they've been conditioned to think white people are more attractive and don't want to be in bodies they would view as less attractive. I could imagine a whole bunch of white people objecting just to be contrary. But overall I can't think of any really good objections from the white point of view.

I presume that "conditioning" refers to social conditioning, i.e. being told or being subjected to media and insinuation that one ethnic group is more attractive than another.

Other (not mutually exclusive) possibilities:

1. Said aesthetic judgements are not purely due to "conditioning", but are (at least in part) formed by the same mental processes as other aesthetic judgements in general.
2. There are certain modules of the mind that render humans likely to form tribal attachments to their ethnic groups (non-EEA condition - just how the adaptations are expressed today). The in-group/out-group dichotomy influences sincere aesthetic judgements.
3. Humans are on average naturally attracted to somewhat similar-looking mates. This influences aesthetic judgements of other ethnic groups in general. Unpacking "natural", genetic causes might interact with early environment e.g. the ethnicity of the humans to which someone is exposed as an infant. #2 and #3 might be closely related causes.

I don’t see why aesthetics shouldn’t be considered a good reason for objecting to the change, whatever the case may be. I suppose humans might be expected to have second-order preferences in favour of allocating relatively little priority to aesthetic preferences that are merely socially conditioned - perhaps - but I don’t see any reason to assume that this is true of the aesthetic preference in question.

Furthermore, homogenising humanity might be considered an aesthetic disutility independent of any comparison between the aesthetic qualities of different ethnic groups – much in the same way that it is a shame when attractive and unique animal species become extinct. Human ethnic groups differ less than different animal species but as humans, the value that many of us attach to diversity and distinctiveness within the human species is magnified.