I think you are conflating thermal conductivity (how well thermal energy propagates through a material, measured in W/kg*K) and thermal capacity (how much energy a given amount of a material can hold, usually measured in J/kg*K).

Water has high values in both btw. So it takes a lot of energy to heat up and cool down, but it also exchanges energy quickly within itself and to it's surroundings.

Now instinctively I thought that water would have a higher thermal conductivity than ice, because Iglus insulate so well. But snow isn't ice (crazy I know:D) and it turns out ice has about a 4 times higher thermal conductivity than water at 0°C.

Therefore: If you cover the ice in a thin layer of water, this should slow down the melting. But if you put the ice into a decent sized container with water, where the total surface area of the mixture becomes more than 4x larger than the surface area of the ice it should speed up the melting process. This effect should also increase the further along the melting process you are, since the surface area of the ice will decrease (less of it left), where as the surface area of the mixture remains mostly equal (ice has a bit less density I know, but small effect).

So I learnt something: Solids generally conduct heat better than liquids. But the original point, where it depends on the container and it's conductive surface area still mostly remains valid.

I'd definitely be interested in the link because I'd imagine it wholly depends on the container, in which you keep the ice-water mixture. A well insulating container, with a small surface area for the water would probably be worse than having the ice in a sieve over the sink. Where as a wide metal bowl should be better. What matters is how much energy from the surrounding air can be transferred into the ice. Water is a great thermal conductor, so I don't see how it could insulate the ice. If you add energy to a mixture of water and ice it will always melt ice. That's why a mixture of water and ice is always at the melting point (given the energy transfer isn't too fast and there is some agitation).

Anyways that's what I'd expect from the experiments I remember from my physics classes.

I'd put it more akin to a survivor bias. The issue stems not from them only looking for evidence supporting their opinion, but them not encountering a fair sample. They only see the drain cleaner "survivors", never those that got unclogged.