On the contrary, heat pumps are one of the most efficient means of controlling temperature. The benefit is you aren't converting from one form to another, you're simply moving it from one place to another. Yes, it uses (rather than loses, a matter of viewpoint) energy to do so.
I have lived in the tropics and subtropics. One of the reasons they are under-developed compared to the temperate zones is because you simply cannot be consistently productive in those temperatures. It is actually dangerous to do much more than lie around in the shade without temperature controlled environments.
Shiina's point first: nothing is wrong with collecting it, but it is not sufficient to guarantee life of multiple people who are hours or days from help. Minimum to sustain life is about 1 gallon per day per person. Two days without it and you will be a raving maniac who would drink a stranger's blood to live. You can only go about 3 days without water before you are not only ill, but you die. Water is one of the primary determinants of habitability on remote islands, and the vast majority of the Pacific and Caribbean Islands have official water concerns, despite the rain. Water deserves significant planning and redundancy.
Now Bob's point and Shiina's second point: It takes significant storage capacity because rain doesn't happen in the same amounts spread perfectly every day. You have to catch as much as you can when it comes. Storage capacity means weight, and volume that has to come at the expense of buoyancy and volume that could be used for other things.
Passive systems are great for energy efficiency, but you need a reserve capacity and safety factor of 3 times minimum requirement for real long term security, which means active systems that can kick in to make up shortfalls. This includes a duration, not just an amount, i.e. minimum comfortable capacity for x persons for x weeks. If you're not planning in these terms, you're failing to plan.
Islands have different freshwater systems, not all the same. You can have volcanic lakes (filled w rainwater), you can have low-lying areas that impound rainwater, you can have a combination of these and enough cracks in the rock structure to make springs appear in different places.
Low-lying coral islands and atolls tend to have none of that. Partly because, with no igneous, solid, elevated ROCK they don't pose enough of a barrier to passing water-laden wind to force precipitation due to altitude & temperature/pressure differences.
What they do often have is a "freshwater lens" In the case of Tuvalu, famous for suing the Czechs for global warming, their wells now have brackish water, which is often referred to as "seawater contamination". However, the basic hydrology of the way a lens works shows that it is not saltwater "intrusion" "contaminating" the freshwater lens, it is simply that there isn't enough freshwater left to displace the saltwater that otherwise naturally infuses the sandy coral soils.
A lens requires some kind of barrier to mixing, and will accumulate as long as withdrawals (human uses, evaporation etc.) don't exceed the rate at which the freshwater is supplied rain, minus the edge mixing with seawater. It's a slow-ish process. It took ~160 years to turn the ~11sq km of Clipperton Island's saltwater lagoon into a freshwater lagoon (for the top 20 meters or so) through accumulation of a lens. Of course, there is more mixing when rain is landing on a trappe dbody of salwater than when it lands on seawater-soaked sand and soil.
I think it is feasible to create an artificial lens on a relatively small scale for fun and profit. If you had a thick-walled tank submersed in seawater, even of a relatively high porosity concrete, and you kept it filled with freshwater, likelihood of "seawater contamination" is very low as long as the basic integrity of the vessel is intact. Micro cracks = no big deal. Macro cracks or holes would be a problem.
The thing is osmosis works in one direction at normal pressures. Less dense freshwater "tries to" dilute denser seawater. Not the other way around. So migration is outward from the vessel to the salt, not from the outside into the fresh. You might "lose" a tiny percentage of your freshwater to osmotic diffusion through the semi-porous container to the sea.