Does the type of strainer built in a teapot matter? I have been making tea in a simple glass jug for several years, but I happened to break it. When I went shopping, I saw many teapots with integrated strainers, which look convenient. 

But there was a large difference between the strainers. Some were made from a fine stainless steel mesh, like a normal strainer. Others were from plastic or glass. The ones from plastic and glass typically had only a few holes, and they were rather small ones. Several had slits instead of round holes. 

Does the type of strainer matter for the quality of the tea? Do a few slits work as well as an all-mesh strainer? Does the water move in and out of the strainer enough to make good tea in a strainer with a few small holes? 

Second, what about the maintenance? Are the slit strainers easier to clean than the mesh ones? 

By the way, the teapots with steel mesh strainers seem to be generally more expensive. Is this because they make better tea, or is it just because metal is more expensive and/or looks better?
I have used several types of integrated strainers and in my experience nothing is better than a wire mesh. The holes and slits in glass and plastic strainers are too small and too few to allow good water circulation, so you have to move the strainer around to get the tea to brew. Cleaning-wise, slits are the worst as their tapered ends trap leaved which can be annoying to remove. The plastic strainers are probably the easiest to clean, with wire being in the middle. 

I'd personally go with the wire mesh type as it brews the best tea.