Yes, when the "thru" outputs of the hardware interface are connected to the line inputs on the mixer your CD players will play thru the hardware interface and into your mixer :-)

I normally hook my CD players to Inputs 1 & 2 on the SSL interface and hook up the Line Outputs to my mixer inputs. So, the one change I need to make is change the RCA from my mixer inputs to the Thru Outputs from the interface. Am I correct?

You'll need to have 4 line inputs connected to play your music from Scratch LINE and directly from your CD players without unplugging anything. 1 pair for the line outs of the interface and the other pair for the thru outputs of the interface.