If you examine the line carefully:
“Arfour, scramble code five to Coruscant, care of the Old Folks’ Home.”
He’s calling himself the Old Folks’ Home, he doesn’t want the Confederacy or any other nosey snoopers knowing that it’s him sending a message to the temple.
But that’s not what “care of” means…?
It means you’re sending a message to someone via someone else. For example, I don’t have my cousin Bob’s address but I know he sees his mom (my aunt) once a week. If I have her address, I could mail a card to “Bob, C/O Aunt Sue, at Aunt Sue’s address”
I think he is indeed using “old folks home” as a code for Jedi Temple, not himself. And that makes more sense anyway. Why would he refer to himself as an institution?
Fun aside; I never even realized that was a line in the film. I think I’ve only watched AOTC 4 times (twice in the theater, once on dvd, and once on Disney+) which is by far the least number of times I’ve watched any Star Wars movie except TROS (but that’s only b/c TROS has been out for so much less time)