For the love of goodness, if I'm having a heart attack or something, please do not waste 5 to 7 seconds total of my life to screw around with footgear instead of just coming inside and getting me. I promise I'd vastly rather have your shoes touch my carpet than die because of "manners".

No. They don't. Every day I see huge numbers of people in lace up oxfords, high heels, and more. And no, people don't ruin their expensive shoes by crushing the heel cups. People do that for cheap shoes, sure, and professionals often wear slip on / slip off shoes intentionally, but it's by no means generalizable.