This reminds me of the predator hierarchy (for example, see Colinvaux's "Why Big Fierce Animals are Rare"): the reinsurers spread the risk from various insurers and for various catastrophes around among a pool of meta-insurers. But this pool is necessarily smaller than that of primary insurers, and their risks more likely to be correlated (catastrophes can cause other catastrophes, and multiply primary insurers can be affected by the same catastrophe).
For that matter, I'm also reminded of credit default swaps, and Lehrer's "We Will All Go Together When We Go."
It's correct that the number of reinsurers is smaller than that of primary insurers. But the risk born by reinsurers is less correlated, not more. Any given primary insurer has risk clusters (domestic market, line of business, etc.). If a large catastrophe happens in their domestic market they might go bust but what are the chances that it happens simultaneously to all markets globally?
Say you're a primary home insurer in the US. If a hurricane hits you might not have enough capital to rebuild all the homes. A reinsurer which is also covering Europe, Asia, LatAm, etc. is less likely to go bankrupt. The reinsurer can cross-subsidize and use the insurance premiums from other regions to pay out the claims from the US market. All that matters is that on average the loss probabilities and severities are estimated correctly.
And this is just using one line of business as example, reinsurers are covering property, casualty, life and health which add extra layers of diversification.
Reinsurance does not only spread risk by pooling multiple insurers, but also smears out the impact of catastrophes geographically and temporally: big events in one year, in one part of the world result in more expensive reinsurance all over the globe for a few years forward, as reinsurers collectively stock up on capital again.
So while locally catastrophes can cause other catastrophes, for the most part earthquakes in Thailand does not trigger wildfires in Texas. Nor does a hurricane in Florida one year cause more hurricanes in Florida the next year.
One aspect worth pointing out is that ILS are transferring insurance risk outside of the insurance industry. Appetite has gone up and down but e.g. hedge funds would normally not be available to assume insurance risk otherwise.