I've only got elite status with AA, and the card stays in my wallet to use at the self-service checkin machines and on the rare occasions I've needed it in-transit (free nuts on AE although usually you don't need to flash the card or BP, once or twice where CX/JL has not recognized the OW Sapphire on the BP when I tried to board for coach on the J line)

For the hotel programs, I just have a bunch of text files in a "reference" folder on my laptop and the SD card on my phone. Ditto for how I keep track of phone #s I don't use enough to keep on my phone's contact list.