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The size of the detector can be very large, stable, and protected with an ice cap. https://icecube.wisc.edu/science/icecube/

There aren't a lot of places with multiple km of water without things like animal life or other confounders.

So, [controlled] liquid water is better but [controlled] solid water is more abundant?

I wouldn't say liquid is "better". The neutrinos don't care from a cross section standpoint.

Uniformity of the light field is going to be different, but that is not my sub-domain.

A liquid you control (and can densely instrument) is going to be a much easier to characterize detector than large volumes of natural material

Wouldn't the higher density of liquid water be an advantage?

It's quite a minor difference

Depends on the energy scale involved!

Higher energy = "easier" to detect (produce more light or radio emission), but the events are rarer so you want to build a bigger detector.

There are also underwater pools of water being used :) (KM3Net,P-ONE, Baikal-GVD, etc.)

For radio detection, solid water is better because it has a much much longer radio attenuation length.