Depends, do you want to live when that unexpected wave comes along?
If your outboard is EFI and starts every single time sure, if it is prone to slow starts, stalling, puking, leave it running. You'll never need it until you really need it
Great fisherman, Charley Soares always leaves his engine on and he does just fine bringing in the big girls. If it is FAC I can get it and wouldn't hesitate to shut it off, but you never know when you may have to move in a hurry. I have caught many fish, big and small with the engine on
Don't know about plugging but with live lining, chunking and throwing eels, I always leave it on. The few times I shut the motor off, I didn't notice a difference in hookup rate. Hell we use to troll pogies at 3kts and would crush large doing it.