Let's say initially every radioactive element was "exploded" into existence from pre-existent elements.
None of these early faster half-lives would be the same as they are today.
As time progressed each would begin to acquire its slower modern-day stable half-life, but would they all acquire these stable rates in a uniformity which would keep them all in synchrony? If they did, all would give the same ages, you are right.
Since carbon dating measures the amount of carbon still in a fossil, then the date given is not accurate.Look at biological breakdown everywhere, it proceeds at different rates.