Is there any advantage or purpose to keeping the engine 'beauty cover' installed? I've heard they just trap heat.

A quick Google search of "what is the purpose of the cover on the engine" returned the following result from https://partsavatar.ca/importance-of-engine-covers "Made from plastic, these engine covers are decent insulators of sound and make the car a little quieter. They also protect from other elements like dust, sand, and water and prevent them from entering every nook and cranny of the engine, like the fuel injectors or the air intake."

Leave it be. It does not trap heat. Removing it will do more harm than good.

By the way ... Your throttle cable looks like it is too loose.

Thanks, I thought the Pathfinder engine cover was so meager as to be purely cosmetic.
Reading the article I laughed about “…invite a rat infestation” since when I popped the hood during first car inspection the owner and I saw rat droppings on top of the intake manifold and gnawed away sections of the hood insulator/blanket (they didn’t drive car much anymore). A quick power wash and trip to junk yard to replace insulator took care of that and no other damage found anywhere else.
Maybe I’ll see how much hood clearance there is and attach a little nut & bolt parts tray to it to make it more useful - lol.

(Slack in throttle cable: old photo I already had on phone).