Why do shelters use clay litter?

Cost savings, easy to purchase and it's what's most often received in the donation box.
I've never had a problem switching litter for my kitties. I know some kitties have their preferences, but in my experience if you introduce the new litter slowly, they'll do all right.

Many do not. More than a decade ago, I worked for what is now the nation's largest cats-only no kill sanctuary. They used a generic version of Feline Pine (they bought white pine wood stove pellets, $4 for 40 lbs). Cheaper than any clay litter, and better at absorbing odors. Six years ago, when I went to work for a public shelter, I got them to convert, too. Now, many area shelters are following suit.

It is a cost saving measure.

It most likely a cost issue but also maybe a cleanliness issue. When you use scoopable litter you aren't always dumping and sanitizing boxes. I am not a shelter but I do foster and I have a litter of kittens now. I don't use much clay litter in box. Clumping kinds need to be at least and inch or 2 deep to work properly. That is a lot of litter to be dumping.

I use just a little clay litter and I just dump the whole box in the morning, wash and sanitize, dry and refill.

In the long run they probably would waste a lot of litter if using clumping kinds. In a shelter setting cleanliness and not spreading disease is a big concern as well.

it's because of the limited budget.