A cooling system is no better than the weakest link, but, ultimately, the only way to get rid of heat from the engine is to transfer the heat from the radiator to the air and that requires moving lots of air THROUGH, not around, the radiator. Since your problem occurs when the car is not moving, you need to move more air through the radiator under those conditions and that means some sort of fan(s).
Closing off the horn holes may be a good idea, but you still need to move lots of air through the radiator. Not knocking electric fans, but even the largest cannot match the air movement of a properly shrouded water pump mounted fan. It may take several HP to turn a water pump mounted fan, but a one HP electric fan would pull something like 60 amps (do the math; 1 HP = 746 Watts, a 1 HP electric motor with 90% efficiency = 829 Watts, 829 Watts @ 13.8 Volts = 60 Amps).
A good water pump mounted fan is a start, but an un-shrouded fan is inherently inefficient. A well fitting shroud will not only improve the fans efficiency, it will also reduce air recirculation. IMO, the Tiger shroud leaves a lot to be desired; the opening is too big for the fan and the bottom of the shroud is open.
If anyone has ever found an alternative shroud that fits, they have done a good job of keeping it a secret. The hardest part of making a shroud is the round opening; ideally, it should have a “tubular” section that completely surrounds the fan with minimal clearance (the less the better, but allow for engine movement). I recommending finding a shroud with the right size opening and then using flat sheetmetal to adapt the center of the shroud to the Tiger radiator.
Just my opinion; your mileage may vary.
PS – My post took a couple of hours from start to finish and I never saw Rob’s post until after the fact. The bottom line is that I don’t want anyone to think I was disagreeing with what he said. Blocking off the horn holes probably reduces recirculation, but a good shroud addresses that issue. As Rob noted, heat buildup in the Tiger engine compartment is an issue and reducing airflow does not seem like a good idea.