One reason to not leave a laptop plugged in is if the battery charger keeps trying to charge battery when full (or dead) it becomes a big fire risk (applies to all devices with battery charging, especially lithium based batteries) .
I haven't heard of issues running laptop without battery, I have seen advice to actually take battery out if not using it as heat from laptop could age battery - I used to run laptops without battery (charging them periodically) - I can see a couple of reasons why you might want to keep battery it - one is poor PSUs may have relied on battery to provide stabilizing (unlikely in laptop) or the battery case is actually important to the structural rigidity of the laptop or required to keep an air gap under case etc.
If it was me I'd run it without battery
Ken
Nostalgia is memory with the pain removed