It is mind blowing to me that the shoes-off-inside rule is so deeply entrenched that emergency responders remove their shoes at the front door
Yes, I think that it's a good example where politeness and respect ingrained in society can be taken a step too far. I don't want my rescuers waste 30-45s on taking shoes off and after I had a stroke and minutes decide whether or not I ended up with a permanent brain damage...
There's photos in the article, those aren't dress shoes with laces to be done and undone, they're fit for the purpose.
Also in cases where it's really dire they'll just step in with shoes on, it's just extremely rare and it might actually be another team altogether, operating under a different protocol.