We've been using teapots for thousands of years.
No kidding, thousands!
Teapots were invented back in the Yuan Dynasty (in China). The design likely progressed gradually from ceramic kettles and wine pots made of metals. Prior to that, only cauldrons were used to boil tea which was then served in bowls.
By the Ming Dynasty, use of the teapot was widespread in China
Early teapots were small in comparison to what we use today, as they were usually designed for a single tea drinker. Believe it or not, once the tea was brewed, they usually drank it straight from the teapot spout! If I'd done that as a child, my grandmother would have clipped my ear! Different strokes for different folks, indeed.
They might have actually been onto something, as single portions of tea are easier to control with regard to flavour and are easier to repeat consistently.
From the 17th century onward, tea was shipped from China to Europe, along with exotic spices and other luxuries. Porcelain teapots, often painted in the familiar blue and white we associate with many Chinese ceramics, were also shipped out.
Here's a fun fact! Because porcelain is completely vitrified, it can stand subjection to seawater without harm. Therefore the teapots could be stowed below deck. The tea, however, had to be stowed above deck in order to remain dry.