Not at all. It's not uncommon to have a fresh water table beneath the surface of an area dotted with islands. If you doubt this, ask yourself where the tree's roots go, and how those salt-intolerant trees survive.
I live 100 feet from salt water in Washington State and I have a perfectly reliable fresh-water well.
See, the mainland has sweetwater running under it all the way to the sea shore and even a little bit beyond, depth and density will vary but it is always there. But a small island surrounded by ocean need not be like that, more than likely it is of a rocky composition (otherwise it would simply disappear or move around) and the water runs right off before it can pool in quantity enough that you could stick a well-head into it to reliably pump it to the surface for consumption.
In such a situation catching it as it falls is your best bet (also better because a small island likely does not have a good filtering capacity and the water you pump up, assuming you can get at it in a useful quantities, will need a lot of processing before you can use it).
It all depends on the actual situation but the chances are rather against it.
Not really. It's a matter of drilling carefully:
http://geography.about.com/library/misc/ucghyben.htm
"On low, small islands that are largely composed of coral or other porous materials, salt water intrusion into the underlying interior is quite common. The drilling or digging of wells on these islands and especially on along the shoreline must be done with care [emphasis added]."
Notice that the author didn't say "pointless."
Most reasonably-sized islands have a subsurface fresh water source. The presence of salt-intolerant trees tells you there's fresh water available beneath the surface.
> ... assuming you can get at it in a useful quantities, will need a lot of processing before you can use it.
If that were true, it would kill the trees. I know a number of locations in Alaska near salt water where, because of geological changes resulting from the 1964 earthquake, even though the surface is dry there's no subsurface fresh water at all, and all the trees promptly died. Those locations are rare.
It's not nearly as simple as drilling a well far from salt water, but it can be done.
