The engine should be "smooth" across the rev range. That doesn't mean there won't be any vibration or noise, as in a Lexus... which are known for "smoothness". Lexus typically isolates their engines very well.

In my experience, the properly running AJ16 will still show a slight "shake" at idle in Park. It might just barely be felt in the cabin. However in Drive, it should be dead smooth and quiet when stopped at idle. On the road, the AJ16 puts out a pleasant but noticeable growl under acceleration, coming to the point of "loudness" at the higher revs. The power delivery should be smooth. But compared to a Lexus, the Jag might seem "noisy" or maybe "harsh" as you say.

Of course, the age of the car matters. Our Jags are getting older and rubbers are deteriorating everywhere, including the engine and transmission mounts. That dry, hard rubber will transmit more engine sound and vibration into the cabin, particularly from the transmission mount. Plus the AJ16 is sensitive to the ignition coils and improper ignition will cause vibration whenever it happens.

I prefer to hear some engine sounds from my car. I don't care for the muted and almost "invisible" engines so many automakers have decided that people prefer. Like Lexus?

I'd say a new transmission mount would be the best step towards reducing engine noise. I have that job planned for my XJ12, as I notice too much engine "drone" on the highway. It's worse than it used to be and I know the rubbers are flattened to nothing.