So, Chase sapphire reserve benefits list that trip delay covers family members even if you aren't traveling with them, as long as they're a family member (long list of specifics - niece/nephew, sister, brother, etc) and ticket charged to card. It doesn't say "has to be in the same household".
If said family member doesn't have the same name, is not at your address, and you weren't traveling with them, how are you supposed to prove family membership to them? Has this ever come up for anyone?
Sorry to hear, if something happened on a trip; hope you recover soon, regardless.
I guess this particular issue comes up (if like the claims portal software flags it), but I haven't dealt with it first-hand yet. If they really need to verify, maybe expect to have to submit birth or marriage certificates, up, down, and/or across lineage to prove 'family.' (Bring out the ole family tree if you have to! Bah!)
Keep in mind, CSR (and most premium cards) insurance coverage is extremely limited. Better protection if you purchase standalone comprehensive insurance policies for each trip and specifically list the name/age of those you wish to cover.