Doormats remove mud but not bacteria. Why would you walk on dirty pavements (where animals wee and poo, people spit, pigeons get squashed etc) and then smear the germs all over your house? It's like rubbing your hands around the toilet bowl, wiping them on a tea-towel then touching all your furniture.
Even if you don't sit on your floors or have children who play on them, why not keep them hygenic? I don't understand the mentality that floors should be as dirty/germy as the pavement outside. They are part of your home 
What if you drop something on a floor where people wear shoes?
Is it laziness that prevents people changing footwear at the front door? I genuinely don't understand why you wouldn't take such a simple, easy step to keep your floors clean.
Most other countries remove shoes at the door. It's very disrespectful not to do this.
I sit on the floor a lot. DS plays on the floor and often drops his toys. We have delicate silk carpets that aren't designed for outdoor shoes.
And yes, at parties everyone removes their shoes (not that they are glamorous parties just big gatherings of friends). Some wear slippers, some go around in socks, barefoot, tights, a few bring soft slip-on shoes for indoor use. If someone refused to go shoe-less I wouldn't let them past the cloakroom/hall because it says they don't respect my home, furniture, cultural background or house-rules.
If you have manky feet, wear socks or slippers (or bring your own indoor footwear). If I go to someone's house I always make sure I have clean deodorised feet and matching socks. If I plan to go barefoot I make sure toenails are painted and manicured.
It's not unwelcoming at all to ask guests to remove shoes, really you shouldn't have to ask. In most parts of the world everyone does it automatically so nobody has to ask and there is no awkwardness!