Don't most people just throw the lid away when the install the filter, for the sound effects?

They might, but then the engine is sucking hot under hood temps.
It's a known fact that a noisy intake makes more power.
I read that in the National Enquirer.

So it's not the chrome?
Is the intercooler so inefficient that a few degrees higher intake temp makes a difference? I don't know. There must be a cheap and easy way to cool the intercooler more.

The more I think. Ambient air gets filtered then heated up by the compressor (probably by 100F or more) then cooled by the intercooler. How is ambient air temp going to have any influence on the temp of the combustion air?

It doesn’t. Considering the intake hose draws air from the cavity in the fender behind the headlight and above the wheel it’s not really cold air intake , the mix of air then passed through the compressor is still warmer even after the intercooler.
its the sounds that are sought after, like flipping the air filter lid on that 4 bbl carb !