Asking for shoes to be removed at the door, does introduce a problem for Carers or any other tradeperson, one of health and safety, even more so for those who are required to wear steel toed safely shoes.
Do they risk bare/socked feet, having anything dropped on to toes, even a tin of talcum power will hurt!
Walking across a carpet that is wet, (tea or wee?)
Plastic slip-on shoe covers, in my experience introduce a moveable layer between the fixed shoe sole and fixed carpet.
As a Carer, bent over the client, the last thing you need is for your foot/feet to slip insde the shoe cover.
Messy shoes are easy to deal with, scrap off under tap if needs be.
Messy socks can only be changed. (do you keep spare socks with you?)
Messy bare feet, how popular are you going to be tramping half way around the house to get to a tap?
An answer to the OP might be IF you have small number of regular Carers then to keep correctly sized indoor shoes available, that can be changed at the door.
Bod.