Having the index's segments file name returned by CoreAdminHandler STATUS can be useful. The info I'm thinking about is the segments file name and its size. If you record that from time to time, in a case of crisis, when u need to restore the index and may not be sure which copy you need to restore, this tiny piece of info can be very useful, as the segmentsN file records the commit point, and therefore what you core reported and what you see at hand can help you make a safer decision.
I also think it's useful info in general, e.g. probably even more than 'version', and it doesn't add much complexity to the handler or the response.