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His claim is that it's necessary - not that it's sufficient.

Open source doesn't mean it HAS been reviewed, it just means it CAN be reviewed. The same can not be said of closed source products. I rather trust the product that at least has the option of being reviewed by an independent source over one that you have no choice but to trust the creator of.

The reason you are able to use those as examples is because they were caught and mitigated.

Open source doesn't magically prevent bugs. But it does ensure that when a bug is found, you'll hear about it and get a well-vetted fix pretty quickly.