What about NHL-sized ice arena using TEC? What's the theoretical limits/bottlenecks here? (asking for a friend)

Heat loss due to resistive heating of the thermoelectric elements is a factor.

I do some work using TECs for cooling optical sensors. Most of the industry has settled on a very small range of materials, such as BiTe, meaning that we're probably at a plateau until a more efficient material comes along. Some of my applications are battery powered, and cooling consumes a fair portion of the total power budget, so a more efficient cooler would be welcome.

Also, TECs are mechanically delicate. There's a laundry list of needs such as being able to form the material into the desired shapes, and make a good mechanical and electrical connection to it, which is usually done with some kind of low melting solder.

It'll be ludicrously expensive to built, and then you'll also have a ridiculous amount of heat to get rid of. The main challenge will be keeping that from melting the ice again (this, in general, is the underlying issue with TECs: they don't have any means of stopping heat from flowing back through them, so it's a bit like trying to bail out a leaking boat, and it gets worse the larger the temperature difference you're trying to achieve. TECs max out their efficiency at a mediocre COP at very low temperature differences, in comparison to compressor based systems which can achieve better COP with more than 10x the temperature difference)