Passive immobilisation also will kick in after the time out period if the vehicle has been running, shut the engine off, and then open the drivers door.
If the passive immobiliser is enabled, then in theory the vehicle will recognise when the key it put in the ignition and the BECM will then activate the coil around the ignition, which pulses the key fob and tells it to then transmit an unlock code - seamlessly letting you start the vehicle again.
However, if you have a fault with the coil, or the key fob then it doesn't send the unlock code, and that's what causes the "Engine Disabled, Press Remote Or Enter Code" message. In a case where you have a working fob but the coil is faulty (as I had on my RR until I turned off the passive immobiliser) then pressing the unlock button on a synced fob will then trigger another unlock and turn the immobiliser off and let you start the vehicle. In a case where the fob is faulty (ie, in your case) then you can't press the button on the remote, as it isn't synced or working properly, so you have to enter the EKA.
Switching the passive immobiliser off - as Gilbertd mentions disables that auto timeout of re-immobilisation, so once it's unlocked, its mobilised and stays that way until the vehicle is re-locked.
As another note - if you get the fob repaired, then you DON'T need to re enable the passive immobiliser for the remote to work. I've had it turned off on mine for the last 5 years and both my remote fobs will sync up and work with no problems. There have been a few noted cases where fobs that won't sync with the vehicle (when they are known to be good) have been made possible to sync by temporarily turning the passive immobiliser back on - but there is no solid evidence as to why this works. I suspect it might have something to do with the RF memory in the BECM, and wonder if just clearing that will have the same affect - but haven't had time, or a spare vehicle with a fob that won't sync, to try it on.
But even if you have a vehicle with a fob that's already synced and turn the passive immobiliser off, it will still stay synced to the vehicle and work normally.
'06 4.2 Supercharged Black/Cream "Jess" -- Bought April 2021 (NZ)
'01 4.6 Vogue Black/Lightstone "Snog" -- Bought Jan 2012
'99 4.6 HSE Cobar Blue/Lightstone -- Sold March 2009
'95 4.6 HSE Epsom Green/Tan -- Traded June 2008