Nobody "rediscovered" paracetamol - its side effects have been known for a long time. Doctors just remind people about them periodically because people treat it like a cure-all and take it like candy. The main thing is: it's toxic to your liver in overdose, and that damage builds up. If you take it rarely for a headache, there's nothing to worry about. But when someone pops a pill for any discomfort several times a day - that's already a problem.
Second: paracetamol masks pain but doesn't treat its cause. You took a pill, the pain went away, but if it's, say, a symptom of something serious, you might just miss it and delay getting diagnosed. Plus it's not that effective anyway - it doesn't help with inflammation unlike other painkillers, it just dulls the pain and fever a bit.
Third - this is more about long-term use: if you take it regularly for months, you can have problems with your kidneys and liver. But again - occasional use for acute headaches within your daily dose is usually safe. The important thing is not to mix it with other medicines that already contain it (a lot of pills do), and not to exceed the dose.