To my understanding, some (but not many) physicists speculate that the Island of stability may contain long-lived elements, as in a billion or so years. But couldn't we rule that out just by the nonexistence of such elements in Nature?
Because no rational process can make them. I've been over the tables, and there are only a couple of possible reactions to get there for any nuclei, and they require two rare ones. Alpha particle capture just isn't going to cut it. Look at the curve; you need more neutrons.
We remember that all elements heavier than iron have primary sources as neutron star collisions and supernovae. The s-process stops around lead or gold (depending on who you ask) leaving the r-process to extend to higher levels. However, the r-process itself is only so rapid and must be bounded by the nuclear decay rates themselves. In order to reach the island of stability by the r process, it would be necessary to pass through the boundary layer around element 109 where several isotopes must be passed through in a row that possess half lives measured in seconds to milliseconds. However my tables show it doesn't even get here but caps out at Neptunium*. Either way, this leaves induced fusion as the only possible pathway.
Unfortunately the induced fusion reactions require rare nuclei. You don't get much chance of a concentrating event of certain elements and try again. Yeah sure you could have a planet collide with a giant star a few hours before said supernova; however consider the odds of getting the result of that exceedingly rare event (consider the timescale) in the gasses required to form our own solar system.
So no, not expected at all.
*One of the Plutoniums (244) has enough half life long enough to last until it gets here, and it's not found that way either, but by direct production in Uranium ores.
Producing ultra-heavy elements in nature is not easy. So their absence "in nature" does not mean they cannot exist or cannot be created given the right conditions.
The valley of stability becomes increasingly n-rich, so neutron capture reactions are essential.
To get beyond lead requires rapid neutron capture in the r-process. The requirements here are a dense flux of neutrons and a capture timescale that is shorter than the beta decay timescale trying to take the nuclei back towards the stability valley.
Once the neutron flux diminishes (these things happen in explosive events like supernovae and neutron star mergers) then beta decay does dominate and takes the nuclei back to the stability valley.
In principle, stable elements of any atomic number could be formed in this way, but ultimately you have to compare neutron capture rates with all the processes that act to destroy the intermediate neutron rich nuclei, such as photodisintegration and fission. For the really heavy nuclei the principle problem is fission - the n-rich heavy nuclei just break up before they can capture any more neutrons given the neutron fluxes that exist in "natural sources".