Riddler's Plan of Getting the Rat in the Light *SPOILERS*

I haven't seen this movie but this sure sounds like a plot hole

Yep

For some reason I thought it was the Penguin who shot him like Jack Ruby did to Lee Harvey Oswald right before they saw the Riddler.

I think they were already on to Falcone based on the Alfred hospital scene. Batman was already suspecting that Falcone had killed Batman's father because of the whole reporter business. Batman learns of this reporter business from Riddler. And there's the hint of "rat with wings" in Spanish. I think Riddler suspected that Batman and Gordon would eventually deduce that Falcone = El rata, even without the recording that Selena gets

The recording simply speeded-up the process.

As detectives, someday, Batman & Gordon would have eventually found out about the Rat with evidence - This is what we have to believe and what Riddler believed.

Btw, major contribution of solving the case was done by Catwoman only. She discovered the dead body of Annika, she found out that it was Kenzie who killed her, she was the one who found and captured Kenzie and finally she only discovered the voice message.

Batman and Gordon just happens to flow through the plot alongside. Well, this is I'd say a characteristic of Film-Noir.