Aga user here. Kettle is filled with cold water, placed on the appropriate hot ‘pad’ on aga, left to get on with heating up while other tasks are done. One big advantage of an aga is that by moving it over slightly, the kettle can be left just staying hot and ready to use for multiple people and mugs. Useful when lots of people are coming in and out.
The teapot holds the teabags or loose tea to which water is added. The kettle never has tea put in it!!! Teabags (an American invention I believe) are extremely convenient for mugs but until the 1960s or so loose tea in a pot would have been used. Tea still out sells coffee in UK and Ireland
A tea cozy is a very useful thing on a big pot. I have a nice knited yellow and black bumble bee cozy for when friends are round and we are having a cuppa. I’m not certain why one isn’t used in the show, particularly if The Green Pot is enamel because those teapots get cold really quickly.
Agas are variable. Oil, solid fuel or electric available, some heat up water for house, some don’t. In our variable climate, somewhere warm and available to dry wet/damp coats, boots, dogs, people at any time is very useful. They are a bit like a beating heart in a house.