If it averages several instances, with enough signal you could decompose a linear combination (e.g. average) of different patterns back out into its constituent parts.
A smarter system won't just take the mean of the votes from different instances but rather discard outliers as invalid input (flagging repeat offenders to be ignored in the future) and use the median or mode of the remainder. The results should also be quantitized to avoid leaking details about sources or internal algorithms; only the larger trends need to be reported.
Of course you could always just keep the collected data private and only provide it to customers willing to pay $$$ for access, which handily limits instance operators' ability to reverse-engineer the source of the data. And nothing prevents you from using separate instances for public and private data sets.
When you have an actual functioning competitive market the money you bring in correlates with the value of the service you provide, so it makes perfect sense to be happy about the money the new surgical center is bringing in. That means it's useful.
The problem is that the health care market is regulated and subsidized in so many ways, many of them conflicting with each other, that competition is very limited and price discovery is reduced to "whatever the patient (and their insurance) can afford to pay" since they can't go anywhere else. Fix that and there won't be any reason for hospital owners or employees to feel guilty about making money.