Airlines send a physical status card to you because it is the ultimate, incontrovertible proof of status.

And if I may turn the question back on you - where does it say that you must collect miles to the account from which you claim status? Please, go and check - but let me tell you that no alliance requires you to collect the miles, to that (or any other) programme, in order to benefit from your status privileges. (Yes, you could insist on flying but without collecting the miles, but you still are entitled your status perks).

Of course, some perks actually require your status number to be in the booking (i.e., if you are entitled to free/extended seating choice prior to OLCI; if your status entitles you to a free upgrade at 48/24 hours prior to departure, etc etc. These type of perks cannot be obtained solely by showing a status card. However, all the main/important ones - free/extra luggage, priority boarding, lounge access etc - only require that you show a valid status fror the alliance on which you are travelling).

I just don't understand why people are apparently so reluctant to carry them; and why people always obsess about having status printed on their boarding pass.

Carry your status card at all times. You cannot be denied your status perks by airline staff if you carry your status card.

Just be watchful and prevent them from overwriting the number in your reservation, if that is what you require.