What do you think is cheaper, a touch display with a a chip that can render graphics etc or a speaker playing an audio stream from a webservice?

From a hobby point of view (meaning that I purchase these things in very small quantities), then cost of both of these is comparable.

I'm ignoring the "touchscreen" part, though, because that's not necessary. I'm thinking just a basic, small TFT display.

Why would you need a touch screen? An 8-segment display with 2 clicker buttons for looping through history, and a little bit of memory (like in a calculator) would be enough.

I think a display might consume much more power than a speak, these look like they could, if they aren't already, powered with batteries, while a display could also be powered with batteries it wouldn't last nearly as long, not to mention that you can hear this even if you're not looking at it.

100%, Many small street vendors don't have a power connection. They literally have a LED light connected to a rechargeable battery at night. I have seen these sound boxes, they probably come with their own chargeable battery. Display cost would be factor too. But most importantly they can service fast while not taking their eyes off, like when they are cooking (for street food vendors)