Returning to the actual subject...
It is possible that no evidence of past bleaching exists because the reef recovered, and the bleached corals are now covered by new healthy growth.
Possible, but extremely unlikely. Even sub-lethal bleaching results in reduced growth in affected corals. That would show up in growth rings, so if mass bleaching were occurring within the past few hundred years we should see several periods where lots of corals have reduced growth concurrently. We don't see that signature in the cores until the past few decades.
Also, if bleaching is severe enough to cause mortality, that gets preserved in the framework of the reef. If you drill into the framework you see mass mortality reflected by changes in species composition and burial rates. Even if the reef subsequently recovers, that mortality is apparent for literally thousands of years. While only a few reefs have been studied so far, we don't see indications of bleaching induced mortality in the past few thousand years.
Also, while there is no doubt that the access to more isolated reefs and better monitoring have been responsible for
some of the increase in reported bleaching events, that doesn't explain it all. Scientists have been studying reefs for over 100 years and have known of bleaching for almost as long. These events are so obvious that you can spot mass bleaching from the surface and even from the air. It's hard to believe that for 70+ years (~20 of which they had SCUBA) they were looking at the reefs but either didn't notice or didn't think to report mass bleaching a single time if the events were occurring as often as now.
The reefs of the world have survived eons of temperature fluctuations, global warming and cooling events, ice ages, etc.. I imagine that they will be here long after we are all gone.
Reefs have been around in some form or another for virtually as long as life on Earth. They have NOT survived continually through eons of fluctuations though. There are multiple periods when the reef-building organisms went extinct or all reef building simply ceased for several million years at a time. The history of reef building is a history of extinction and then the evolution of new reef-building species, not of reef-builders simply persisting regardless of what happens around them.
Modern corals started building reefs back when dinosaurs were around. They subsequently went almost extinct and completely stopped building reefs for several million years. Reefs only re-emerged about 23 million years ago when modern corals started to re-diversify. In those intervening years corals have survived ice ages that were much cooler than modern temperatures and interglacials that were warmer. However, they haven't experienced rates of change comparable to what they're facing now.
If history is any indication, coral reefs can and will go extinct. If we drive that extinction along,
something will eventually come back and build reefs again, but we could be waiting a few million years for that to happen.