In firefighting we sometimes use a tactic called positive pressure attack, where we'll point a very powerful fan into the front door and break the window of the fire room, at which point the fan creates a flow path taking much of the smoke and heat out of the rest of the house. It's pretty dramatic how quickly the conditions can improve inside while the fire is still burning, making the rest of the structure more survivable for victims, much easier for us to operate to find victims and find & put out the fire. It is ultimately making a very very low, but still increased, pressure situation at the front door to force the air/smoke out the lower pressure environment outside that window.
Having a screen on the fire room window cuts that flow path down dramatically. Like 75% reduction or more. Granted, this is a very low pressure scenario, but I'm not sure how much pressure is created at the grille either. Or how much that restricts flow compared to the already restrictive condenser/radiator/etc.
Our cooling systems are very robust.. I don't think having that mesh grille guard would cause any real temperature concerns, but my gut is telling me it will have an impact and the fan is going to have to pick up the slack. Edit: whether this is a significant amount of slack pickup or not probably has to do with other elements of a build.. how much harder is the vehicle having to work to go 75mph from front bumper, rack, tires, etc.