Just got my 2011 Jaguar XJL 5.0 NA for a week and I noticed some things about the car.
Car history: New in 2011 and averaged about 1500-2000 miles/year up until 2019. From 2019 to when I bought it last month the car was only driven about 100 miles/year and was parked in the driveway during this time. PPI showed nothing major, just recommended new tires due to dry rot, coolant and brake fluid flush, and brakes at 20%. But yea.... car was sitting for a solid 8 or so months prior to my purchase, owner replaced battery and did an oil change at the start of this month.

1. Car vibrates on start up & idle, vibrates in N also.
2. I haven't driven another XJ/XJL so I don't know if this is normal or not but the gas pedal is stiff and requires more pressure to push. The cars I've recently driven for an extended period of time are: 2013 Honda CRV, 2018 Honda Civic, 2019 Honda Accord, 2016 BMW X5, 2014 ML350, 2016 Maserati Granturismo and none of their gas pedals are anywhere near as stiff as the Jag's. I'd estimate the Jag's gas pedal is 2-3x more stiff than any of these cars. Not sure if this is a issue but it sure does help with smooth linear acceleration.
3. The brake sometimes feels like its still partially engaged for about 2-3 seconds after I let go and when pressing the gas. I'm not sure if this is an issue with the brake of maybe it might have something to do with sluggish initial acceleration, don't know exact which one I'm feeling.

 

I don't know the answer to your questions. But, I do have a recommendation. Take the car into a good Indie or a dealer and find out if you have any issues.

 

Yea, going to the dealer would be my next step, my local jaguar specialist didn't mention this issue during the PPI so I'm kind of hesitant to go back to them.