I don't think this analogy maps very well.
One difference is the time frame involved. The Fermi Paradox remains very robust because we can view galaxies from Andromeda, which is only a few million light-years away, to the Hubble Ultra Deep Field, which has galaxies billions of light-years away (and therefore viewed billions of years in the past). We have an entire spectrum of time presented to us thanks to varying distances, and we see no evidence of intelligent life anywhere we look.
And near-light travel could settle whole portions of the sky in mere eyeblinks by comparison to how long galaxies exist. Sitting around on an island for 30 to 100 years doesn't compare.
Another difference is the fact that the humans already had a functioning civilization somewhere, and traveled to that island, which would be analogous to humans settling other parts of the galaxy rather than being stuck on a single planet where we still are. The islanders are already past the filter point.
The two situations aren't identical, they just have some similarities when it comes to applying anthropic reasoning. The filter in James' example can be in two possible places: getting to the island, or surviving on the island. These are analogous to the filter being during the rise of civilization and being during civilization surviving long enough to colonize the galaxy.
If our civilization doesn’t collapse then in 50 to 1000 years humanity will almost certainly start colonizing the galaxy. This seems inconsistent with the fact that there are a huge number of other planets in our galaxy but we have not yet found evidence of extraterrestrial life. Drawing from this paradox, Robin Hanson writes that there should be some great filter “between death and expanding lasting life, and humanity faces the ominous question: how far along this filter are we?”
Katja Grace reasons that this filter probably lies in our future and so we are likely doomed. (Hanson agrees with Grace’s argument.) Please read Grace’s post before reading the rest of this post.
Small groups of humans have been in situations similar to that currently faced by our entire species. To see this imagine you live in a prehistoric hunter gatherer tribe of 200 people. Your tribe lives on a large, mostly uninhabited island. Your grandparents, along with a few other people came over to the island about 30 years ago. Since arriving on the island no one in your tribe has encountered any other humans. You figure that if your civilization doesn’t get wiped out then in 100 or so years your people will multiply in number and spread throughout the island. If your civilization does so spread, then any new immigrants to the island would quickly encounter your tribe.
Why, you wonder, have you not seen other groups of humans on your island? You figure that it’s either because your tribe was the first to arrive on the island, or because your island is prone to extinction disasters that periodically wipes out all its human inhabitance. Using anthropic reasoning similar to that used by Katja Grace you postulate the existence of four types of islands:
1) Islands that are easy to reach and are not prone to disaster.
2) Islands that are hard to reach and are not prone to disaster.
3) Islands that are easy to reach and are prone to disaster.
4) Islands that are hard to reach and are prone to disaster.
You conclude that your island almost certainly isn’t type (1) because if it were you would have almost certainly seen other humans on the island. You also figure that on type (3) islands lots of civilizations will start and then be destroyed. Consequently, most of the civilizations that find themselves on types (2), (3) or (4) islands are in fact on type (3) islands. Your tribe, you therefore reason, will probably be wiped out by a disaster.
I believe that the argument for why you are probably on a type (3) island is analogous to that of why the great filter probably lies in our future because both come about from updating “on your own existence by weighting possible worlds as more likely the more observers they contain.”
If, therefore, Grace’s anthropic reasoning is correct then most of the time when a prehistoric group of humans settled a large, uninhabited island, that group went extinct.