You started this thread asking whether someone coming for an appointment was reasonable for not removing their shoes.
We can all agree they should have done a better job of wiping their feet.
You also say that if people had wooden floors it would be different, but you think it's rude for people to walk in with outside shoes on and asked whether you would be unreasonable to ask people to remove them.
"I just think it's gross but maybe I'm the odd one! It's just how I was brought up!"
Surely what this thread has proven is that opinions are divided on this matter. I was brought up to make people feel at home. My Mum was incredibly house proud but would not have dreamt of ever asking people to remove their shoes (and we had a cream carpet) and would have considered it the height of bad manners. It's just the way I was brought up.
So to some people, yes you are being unreasonable to ask people to remove their shoes. If they feel comfortable enough to offer, that's a different story.
Some people will think you are rude to insist, and it will make some people feel unwelcome.
But it strikes me that you didn't start this thread for genuine opinions, you started this thread so that people could pile in agreeing with you that other people are vile and disgusting, and sympathising with you over the mark on your carpet.
Keeping things nice and tidy isn't entitled, but doing it at the expense of other people's quite justified comfort (problem feet, mobility issues etc) is.
You crack on insisting that people take their shoes off. Maybe give them a towel to sit on too. But be aware that some people will think you are rude and be OK with that.