It seems to me that if you know the "specific gene and there are only 34 in the world known to have this", there would be some useful value if 23andme or the other DNA Services tested for that specific gene.
I realize there are a lot if "if's", but if 23andme tests the gene values that are also being tested at another project, then IMO, would be of genealogical interest to be able to trace the history of this particular gene.
You may want to contact 23andme and see if you, or your cousins, qualify for a free kit. I have had a few cousins tell me that they had some rare genetic condition, and the kit's were free. They didn't even know about DNA Relatives. They were surprised when they were first contacted by their DNA Cousins.
If you or anyone needs me to look up a particular gene at 23andme, just contact me and I will look it up and see what it tests for.