Does the type of strainer built in a teapot matter?
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.