It seems that the model years 2000 and 2001 have a lot of transmission and head gasket problems. Both are related to not having fluids changed on a regular basis albeit most cars will run just fine without ever changing/flushing the transmission fluid and engine coolant. But this is a simple and cheap measure if you are aware to do it every 30,000 miles or so, otherwise the consequences can be very costly. By and large though, I agree that most of the people that I know that have Centuries are very pleased with them and have been long-time Buick customers. I know I would not hesitate to buy one, maybe a '00 or '01. The newer ones are next to bullet-proof.
I just overheard my shop mechanic telling a customer that the 3.1 L GM engine is known for lower intake gasket failure. Watch for slow loss of coolant. If the gasket fails and you don't have it fixed the head gasket will fail too.
After 2001 the lower intake manifold gasket design was changed.
I have had 5 Buick LeSabres. With the exception of severe rust problem-covered by GM, I have been satisfied.
As a loyal Buick owner, I purchased a low mileage used 2002 Century for my daughter. It has been a disaster. Two power window lifts, new bearing sensor kits, noisy piston slap at start up. This all happened to a car with 42 k, in the first two months. I will not buy another Buick, nor another GM product based on this experience. GM wouldn't even acknowledge the noisy engine start up as anything but normal, for the engine.