At this point, I'm not sure what you even mean about public health official greenlighting protests. Both protests were allowed to happen. Both had parts of protests that were preplanned and parts that weren't. Are you just talking about the comments Fauci made in an unofficial capacity?

>Even if this were the case (I'd be curious to see cites of actual papers), that indicates a failure on the part of public health authorities: they should have been greenlighting the anti-lockdown protests as consistent with public health, so long participants masked appropriately.

https://www.nber.org/system/files/working_papers/w27408/w274...

>See e.g. https://www.nytimes.com/2020/12/05/health/covid-vaccine-firs... (non-paywall https://indianexpress.com/article/explained/the-elderly-vs-e...). To jump to the bit in question, ctrl-f "Ultimately, the choice comes down to whether preventing death"

That appears to be an argument about saving lives versus returning to normal. They aren't arguing that the vaccine should be given to people from diverse backgrounds just because. They are arguing that essential workers getting vaccinated is more important than getting the elderly vaccinated. They are simply noting the essential workers have a "high proportion of minority, low-income and low-education workers" which means they often aren't valued politically as much as other groups. Either way, they were only talking about recommendations that have no real power in deciding what happens with prioritization.