There is already a very old existing issue: SERVER-12869. The original reporter is already inactive. This issue is an attempt to reach the MongoDB developers and try to stress out that the issue effectively disallows use of index `{$exists: true}` operation. And as such it is a significant bug. Not being able to use index for $exists comes at surprise for developers and many don't even notice an issue until there is enough data, or until explicitly checking the explain() results. Even with explain() the cause of behavior was not understood until I made an assumption that the index doesn't distinguish between nulls and non-existing items and then confirmed that experimentally.
For some reason I cannot tag specific MongoDB developers in this Jira server so have to create the ticket to attract attention to my comments on the mentioned ticket. I've provided the potential (pretty obvious) solution there, and propose to discuss viability in the original ticket.