Advantages of using a non-glass teapot?
It depends on personal tastes (both aesthetic and flavour). Ceramic pots certainly retain heat longer which suits black tea as a pot can be on the go for a while. Clay is for the connoisseur and should be kept for one specific tea. These ideally need to be used regularly so don't suit most households (including me).

I'd personally have:


A western style ceramic pot  (or two, one small, one large) for black teas like Kenyan,
Ceylon or Assam based teas which I'd only rinse out (unless it got forgotten and grew mold! in which case it gets a good scrub) 
A glass pot for strongly flavoured teas or heavy tannin tea like Lapsang Souchong
or chai (if i make it in a pot) which would be cleaned each time 
An eastern style ceramic pot for green tea as they generally keep the larger leaves back.


The biggest difference I've found is that warming the pot is more significant when using ceramic or clay over glass as the material absorbs more heat.