I agree that they're lying, but I wouldn't be surprised if they really do stop training on "live" internet data.
The lie is in their stated reasoning. This has nothing to do with AI being "too dangerous" to release into the wild. There's no reason to involve AI in gathering training data anyway... it's always been possible to scrape the internet quickly and cheaply with non-AI crawlers and bots.
The real reason is that the internet just doesn't have enough useful training data anymore. There's so much AI slop out there that the frontier labs risk model collapse if they continue to train on it. Large swathes of the internet have also become openly hostile to AI, either blocking the traffic outright or redirecting their crawlers into AI tarpits that intentionally waste their time and feed them endless reems of useless garbage. They've poisoned the proverbial well for everyone and now they're afraid to drink the water.